KILLED proof of input_6TycMuKv9h.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 8 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 365 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 61 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 265 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 95 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 448 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 109 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 3507 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1579 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 6 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 126 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 125 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 4 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 113 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 203 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 198 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 143 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [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) CdtRewritingProof [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), 7 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) 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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x), y) -> f(-(*(s(x), s(y)), s(*(s(x), y))), *(y, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) f(s(x), y) -> f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x), y) -> f(-(*(s(x), s(y)), s(*(s(x), y))), *(y, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] +(0, y) -> y [1] +(s(x), y) -> s(+(x, y)) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(x, *(x, y)) [1] f(s(x), y) -> f(-(*(s(x), s(y)), s(*(s(x), y))), *(y, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus + => plus * => times ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x), y) -> f(minus(times(s(x), s(y)), s(times(s(x), y))), times(y, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x), y) -> f(minus(times(s(x), s(y)), s(times(s(x), y))), times(y, y)) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> 0:s -> f Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: f_2 (c) The following functions are completely defined: times_2 minus_2 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x), y) -> f(minus(times(s(x), s(y)), s(times(s(x), y))), times(y, y)) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(0)) -> plus(x, 0) [2] times(x, s(s(y'))) -> plus(x, plus(x, times(x, y'))) [2] f(s(x), 0) -> f(minus(plus(s(x), times(s(x), 0)), s(0)), 0) [4] f(s(x), s(y'')) -> f(minus(plus(s(x), times(s(x), s(y''))), s(plus(s(x), times(s(x), y'')))), plus(s(y''), times(s(y''), y''))) [4] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 const => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + x, times(1 + x, 0)), 1 + 0), 0) :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + x, times(1 + x, 1 + y'')), 1 + plus(1 + x, times(1 + x, y''))), plus(1 + y'', times(1 + y'', y''))) :|: z' = 1 + y'', x >= 0, y'' >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> plus(x, plus(x, times(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x times(z, z') -{ 2 }-> plus(x, 0) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { plus } { times } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z*z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 0)), 1 + 0), 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 4 }-> f(minus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z' - 1))), 1 + plus(1 + (z - 1), times(1 + (z - 1), z' - 1))), plus(1 + (z' - 1), times(1 + (z' - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 13 + s10 + 2*z + 4*z*z' + 12*z' + 2*z'^2 }-> f(s11, s13) :|: s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z' - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s11 >= 0, s11 <= s8, s12 >= 0, s12 <= 1 + (z' - 1) + 2 * ((z' - 1) * (1 + (z' - 1))), s13 >= 0, s13 <= 1 + (z' - 1) + s12, z - 1 >= 0, z' - 1 >= 0 f(z, z') -{ 11 + 2*z }-> f(s6, 0) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * (0 * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 13 + s10 + 2*z + 4*z*z' + 12*z' + 2*z'^2 }-> f(s11, s13) :|: s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z' - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s11 >= 0, s11 <= s8, s12 >= 0, s12 <= 1 + (z' - 1) + 2 * ((z' - 1) * (1 + (z' - 1))), s13 >= 0, s13 <= 1 + (z' - 1) + s12, z - 1 >= 0, z' - 1 >= 0 f(z, z') -{ 11 + 2*z }-> f(s6, 0) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * (0 * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 13 + s10 + 2*z + 4*z*z' + 12*z' + 2*z'^2 }-> f(s11, s13) :|: s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z' - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s11 >= 0, s11 <= s8, s12 >= 0, s12 <= 1 + (z' - 1) + 2 * ((z' - 1) * (1 + (z' - 1))), s13 >= 0, s13 <= 1 + (z' - 1) + s12, z - 1 >= 0, z' - 1 >= 0 f(z, z') -{ 11 + 2*z }-> f(s6, 0) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * (0 * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: INF, size: O(1) [0] ---------------------------------------- (45) 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: minus(v0, v1) -> null_minus [0] f(v0, v1) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_minus, null_f, null_plus, null_times ---------------------------------------- (46) 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x), y) -> f(minus(times(s(x), s(y)), s(times(s(x), y))), times(y, y)) [1] minus(v0, v1) -> null_minus [0] f(v0, v1) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] The TRS has the following type information: minus :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times 0 :: 0:s:null_minus:null_plus:null_times s :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times plus :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times times :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times f :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> null_f null_minus :: 0:s:null_minus:null_plus:null_times null_f :: null_f null_plus :: 0:s:null_minus:null_plus:null_times null_times :: 0:s:null_minus:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (47) 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_minus => 0 null_f => 0 null_plus => 0 null_times => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(minus(times(1 + x, 1 + y), 1 + times(1 + x, y)), times(y, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> plus(x, times(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0), z1) -> f(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)) Tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(0, z0) -> c2 +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, 0) -> c4 *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) S tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(0, z0) -> c2 +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, 0) -> c4 *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, f_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c, c1_1, c2, c3_1, c4, c5_2, c6_3, c7_3, c8_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: +'(0, z0) -> c2 -'(z0, 0) -> c *'(z0, 0) -> c4 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0), z1) -> f(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, f_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c8_2 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0), z1) -> f(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c8_2 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), z1) -> c6(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) by F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c8_2, c6_3 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), z1) -> c7(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), -'(*(s(z0), s(z1)), s(*(s(z0), z1))), *'(s(z0), z1)) by F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c6_3, c7_3 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c6_3, c7_3, c7_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), z1) -> c8(F(-(*(s(z0), s(z1)), s(*(s(z0), z1))), *(z1, z1)), *'(z1, z1)) by F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), *'(0, 0)) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), *'(0, 0)) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), *'(0, 0)) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1, c6_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. F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c7_3, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), z1) -> c6(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), s(z1))) by F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c6(-'(*(s(x0), s(x1)), s(*(s(x0), x1))), *'(s(x0), s(x1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c6(-'(*(s(x0), s(x1)), s(*(s(x0), x1))), *'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c6(-'(*(s(x0), s(x1)), s(*(s(x0), x1))), *'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c6_2 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) by F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) by F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c7(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c7(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c7(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1 ---------------------------------------- (87) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c2_1 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c2_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), z1) -> c7(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), -'(*(s(x0), s(z1)), s(*(s(x0), z1))), *'(s(x0), z1)) by F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), *(s(x0), s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), 0)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) by F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c7_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) by F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) by F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), z1) -> c8(F(-(+(s(x0), *(s(x0), z1)), s(*(s(x0), z1))), *(z1, z1)), *'(z1, z1)) by F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), *(s(x0), s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), *'(0, 0)) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), *'(0, 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), *'(0, 0)) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), *'(0, 0)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(*(s(x0), 0))), 0)) by F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c8_1, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), *(0, 0))) by F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z0)) -> c6(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(s(z0)))) by F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) by F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) by F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) by F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c6(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(x1)) -> c6(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(s(x1)))) by F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(s(z1))) -> c6(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(s(z1))))) by F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(0)) -> c6(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(s(0)))) by F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) by F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) by F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_1, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) by F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(*(s(x0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_2 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(x1)) -> c6(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(s(x1)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples: F(s(x0), s(x1)) -> c6(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), 0) -> c6(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(z0), 0) -> c6(-'(*(s(z0), s(0)), s(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1, c6_2 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c6(F(-(+(s(x0), *(s(x0), s(z1))), s(+(s(x0), *(s(x0), z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1, c6_2 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), x1) -> c6(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), s(x1))) by F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1, c6_2 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), 0) -> c6(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0))), *'(s(x0), s(0))) by F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), -'(*(s(z0), s(0)), s(0)), *'(s(z0), s(0))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1, c6_2 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), *'(s(z0), s(0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), *'(s(z0), s(0))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c6_3, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_1, c1_1, c6_2 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c6(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(z1)) -> c6(F(-(+(s(z0), +(s(z0), *(s(z0), z1))), s(*(s(z0), s(z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(-'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1))))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), +(s(z0), *(s(z0), z1))), s(*(s(z0), s(z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0), s(z0)) -> c7(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), -'(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), *'(s(x0), s(z0))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(z1)) -> c7(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), s(x1)) -> c7(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), -'(*(s(z0), s(s(x1))), s(*(s(z0), s(x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c7(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), *'(s(x0), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), -'(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), *'(s(x0), s(0))) F(s(x0), s(x1)) -> c7(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), -'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1)))), *'(s(x0), s(x1))) F(s(z0), x1) -> c7(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), -'(*(s(z0), s(x1)), s(*(s(z0), x1))), *'(s(z0), x1)) F(s(x0), s(z1)) -> c7(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), -'(*(s(x0), s(s(z1))), s(*(s(x0), s(z1)))), *'(s(x0), s(z1))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0)), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), 0) -> c7(F(-(*(s(x0), s(0)), s(0)), 0), -'(*(s(x0), s(0)), s(*(s(x0), 0)))) F(s(x0), s(z0)) -> c8(F(-(*(s(x0), s(s(z0))), s(*(s(x0), s(z0)))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(*(s(x0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(*(s(x0), s(0)))), +(s(0), 0)), *'(s(0), s(0))) F(s(x0), s(z1)) -> c8(F(-(*(s(x0), s(s(z1))), s(+(s(x0), *(s(x0), z1)))), +(s(z1), *(s(z1), z1))), *'(s(z1), s(z1))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(*(s(x0), s(x1)))), +(s(x1), *(s(x1), x1))), *'(s(x1), s(x1))) F(s(z0), s(x1)) -> c8(F(-(*(s(z0), s(s(x1))), s(s(+(z0, *(s(z0), x1))))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(x0), s(s(z1))) -> c8(F(-(*(s(x0), s(s(s(z1)))), s(+(s(x0), +(s(x0), *(s(x0), z1))))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(*(s(x0), s(s(0))), s(+(s(x0), 0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(x0), s(x1)) -> c8(F(-(+(s(x0), *(s(x0), s(x1))), s(+(s(x0), *(s(x0), x1)))), *(s(x1), s(x1))), *'(s(x1), s(x1))) F(s(z0), x1) -> c8(F(-(s(+(z0, *(s(z0), x1))), s(*(s(z0), x1))), *(x1, x1)), *'(x1, x1)) F(s(x0), s(z1)) -> c8(F(-(+(s(x0), +(s(x0), *(s(x0), z1))), s(*(s(x0), s(z1)))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(*(s(x0), 0))), 0)) F(s(x0), 0) -> c8(F(-(+(s(x0), *(s(x0), 0)), s(0)), *(0, 0))) F(s(x0), 0) -> c8(F(-(+(s(x0), 0), s(*(s(x0), 0))), *(0, 0))) F(s(x0), 0) -> c8(F(-(*(s(x0), s(0)), s(0)), 0)) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(*(s(z0), s(z1)))), s(+(z1, *(s(z1), z1)))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(*(s(z0), s(s(z1))))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(*(s(z0), s(0)))), +(s(0), 0)), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(*(s(z0), s(z1)))), +(s(z1), *(s(z1), z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c6(F(-(*(s(z0), s(s(z1))), s(s(+(z0, *(s(z0), z1))))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) F(s(z0), s(s(z1))) -> c6(F(-(*(s(z0), s(s(s(z1)))), s(+(s(z0), +(s(z0), *(s(z0), z1))))), *(s(s(z1)), s(s(z1)))), -'(*(s(z0), s(s(s(z1)))), s(+(s(z0), *(s(z0), s(z1))))), *'(s(z0), s(s(s(z1))))) F(s(z0), s(0)) -> c6(F(-(*(s(z0), s(s(0))), s(+(s(z0), 0))), *(s(0), s(0))), -'(*(s(z0), s(s(0))), s(+(s(z0), *(s(z0), 0)))), *'(s(z0), s(s(0)))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), *(s(z0), s(z1))), s(+(s(z0), *(s(z0), z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0), 0) -> c6(F(-(*(s(z0), s(0)), s(0)), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(*(s(z0), 0))), 0), *'(s(z0), s(0))) F(s(z0), 0) -> c6(F(-(+(s(z0), *(s(z0), 0)), s(0)), *(0, 0)), *'(s(z0), s(0))) F(s(z0), z1) -> c6(F(-(s(+(z0, *(s(z0), z1))), s(*(s(z0), z1))), *(z1, z1)), -'(+(s(z0), *(s(z0), z1)), s(*(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), 0) -> c6(F(-(+(s(z0), 0), s(*(s(z0), 0))), *(0, 0)), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c6(F(-(+(s(z0), +(s(z0), *(s(z0), z1))), s(*(s(z0), s(z1)))), *(s(z1), s(z1))), -'(*(s(z0), s(s(z1))), s(+(s(z0), *(s(z0), z1)))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), x1) -> c(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), x1) -> c(*'(s(x0), s(x1))) F(s(x0), s(x1)) -> c2(-'(*(s(x0), s(s(x1))), s(*(s(x0), s(x1))))) F(s(x0), s(x1)) -> c2(*'(s(x0), s(x1))) F(s(x0), x1) -> c7(-'(*(s(x0), s(x1)), s(*(s(x0), x1)))) F(s(x0), 0) -> c7(-'(*(s(x0), s(0)), s(*(s(x0), 0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_2, -'_2 Compound Symbols: c3_1, c5_2, c_1, c7_3, c2_1, c7_1, c7_2, c8_2, c8_1, c6_3, c6_1, c1_1, c6_2