KILLED proof of input_KXVZeBZ7Wo.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 265 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 81 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 316 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 99 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 498 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 146 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 2619 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 1275 ms] (52) CpxRNTS (53) CompletionProof [UPPER BOUND(ID), 0 ms] (54) CpxTypedWeightedCompleteTrs (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtUsableRulesProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) 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), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) 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)) p(s(x)) -> x f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) 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)) p(s(x)) -> x f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 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)) p(s(x)) -> x f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) 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] p(s(x)) -> x [1] f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) [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] p(s(x)) -> x [1] f(s(x)) -> f(minus(p(times(s(x), s(x))), times(s(x), s(x)))) [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] p(s(x)) -> x [1] f(s(x)) -> f(minus(p(times(s(x), s(x))), times(s(x), s(x)))) [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 p :: 0:s -> 0:s f :: 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_1 (c) The following functions are completely defined: times_2 minus_2 p_1 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] p(v0) -> 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] p(s(x)) -> x [1] f(s(x)) -> f(minus(p(times(s(x), s(x))), times(s(x), s(x)))) [1] minus(v0, v1) -> 0 [0] p(v0) -> 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 p :: 0:s -> 0:s f :: 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] p(s(x)) -> x [1] f(s(x)) -> f(minus(p(plus(s(x), times(s(x), x))), plus(s(x), times(s(x), x)))) [3] minus(v0, v1) -> 0 [0] p(v0) -> 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 p :: 0:s -> 0:s f :: 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) -{ 3 }-> f(minus(p(plus(1 + x, times(1 + x, x))), plus(1 + x, times(1 + x, x)))) :|: x >= 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + x, times(1 + x, x))), plus(1 + x, times(1 + x, x)))) :|: x >= 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { plus } { p } { times } { f } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} ---------------------------------------- (25) 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 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (27) 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' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (31) 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' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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'] ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {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) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {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'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {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'] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (43) 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' ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (45) 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' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(p(plus(1 + (z - 1), times(1 + (z - 1), z - 1))), plus(1 + (z - 1), times(1 + (z - 1), z - 1)))) :|: 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 7 + s8 + 8*z + 4*z^2 }-> f(s9) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s6, 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (49) 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 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 7 + s8 + 8*z + 4*z^2 }-> f(s9) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s6, 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [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] ---------------------------------------- (51) 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: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 7 + s8 + 8*z + 4*z^2 }-> f(s9) :|: s4 >= 0, s4 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s5 >= 0, s5 <= 1 + (z - 1) + s4, s6 >= 0, s6 <= s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s6, 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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'] p: runtime: O(1) [1], size: O(n^1) [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] ---------------------------------------- (53) 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] p(v0) -> null_p [0] f(v0) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_minus, null_p, null_f, null_plus, null_times ---------------------------------------- (54) 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] p(s(x)) -> x [1] f(s(x)) -> f(minus(p(times(s(x), s(x))), times(s(x), s(x)))) [1] minus(v0, v1) -> null_minus [0] p(v0) -> null_p [0] f(v0) -> 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_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times 0 :: 0:s:null_minus:null_p:null_plus:null_times s :: 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times plus :: 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times times :: 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times p :: 0:s:null_minus:null_p:null_plus:null_times -> 0:s:null_minus:null_p:null_plus:null_times f :: 0:s:null_minus:null_p:null_plus:null_times -> null_f null_minus :: 0:s:null_minus:null_p:null_plus:null_times null_p :: 0:s:null_minus:null_p:null_plus:null_times null_f :: null_f null_plus :: 0:s:null_minus:null_p:null_plus:null_times null_times :: 0:s:null_minus:null_p:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (55) 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_p => 0 null_f => 0 null_plus => 0 null_times => 0 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(minus(p(times(1 + x, 1 + x)), times(1 + x, 1 + x))) :|: x >= 0, z = 1 + x f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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. ---------------------------------------- (57) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (58) 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)) p(s(z0)) -> z0 f(s(z0)) -> f(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))) 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)) P(s(z0)) -> c6 F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), P(*(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) 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)) P(s(z0)) -> c6 F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), P(*(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, p_1, f_1 Defined Pair Symbols: -'_2, +'_2, *'_2, P_1, F_1 Compound Symbols: c, c1_1, c2, c3_1, c4, c5_2, c6, c7_4, c8_3 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: P(s(z0)) -> c6 +'(0, z0) -> c2 -'(z0, 0) -> c *'(z0, 0) -> c4 ---------------------------------------- (60) 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)) p(s(z0)) -> z0 f(s(z0)) -> f(-(p(*(s(z0), s(z0))), *(s(z0), s(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)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), P(*(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) 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)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), P(*(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, p_1, f_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c7_4, c8_3 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) 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)) p(s(z0)) -> z0 f(s(z0)) -> f(-(p(*(s(z0), s(z0))), *(s(z0), s(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)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) 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)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, p_1, f_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (63) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0)) -> f(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) 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)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), 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(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c8_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), 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(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c8(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), 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(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), 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(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c8_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c7(F(-(p(*(s(z1), s(z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), 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(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c7_3, c8_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), *(s(z1), s(z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), 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(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) 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(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c8(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) by F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) 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(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) by F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(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(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) 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)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c8(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) 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)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> 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(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) 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(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c8_3, c7_3 ---------------------------------------- (89) 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))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(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(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: +'_2, *'_2, F_1, -'_2 Compound Symbols: c3_1, c5_2, c8_3, c7_3, c1_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) by F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: +'_2, *'_2, F_1, -'_2 Compound Symbols: c3_1, c5_2, c8_3, c7_3, c1_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c8(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) by F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: +'_2, *'_2, F_1, -'_2 Compound Symbols: c3_1, c5_2, c7_3, c8_3, c1_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: +'_2, *'_2, F_1, -'_2 Compound Symbols: c3_1, c5_2, c7_3, c8_3, c1_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(z0), z1) -> c3(+'(z0, z1)) by +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c7(F(-(p(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) by F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) by F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c7(F(-(p(+(s(z1), *(s(z1), z1))), +(s(z1), *(s(z1), z1)))), -'(p(*(s(z1), s(z1))), *(s(z1), s(z1))), *'(s(z1), s(z1))) by F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), *(s(z0), s(z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), *(s(0), s(0))), *'(s(0), s(0))) by F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c7(F(-(p(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), *(s(s(z1)), s(s(z1))))), -'(p(*(s(s(z1)), s(s(z1)))), *(s(s(z1)), s(s(z1)))), *'(s(s(z1)), s(s(z1)))) by F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) by F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) by F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c1_1, c7_3, c3_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, F_1, +'_2 Compound Symbols: c5_2, c1_1, c8_3, c7_3, c3_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) by F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, F_1, +'_2 Compound Symbols: c5_2, c1_1, c8_3, c7_3, c3_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) by F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, F_1, +'_2 Compound Symbols: c5_2, c1_1, c7_3, c3_1, c8_3 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) by F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) by F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), +(s(z0), *(s(z0), z0))), *'(s(z0), s(z0))) by F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), +(s(0), *(s(0), 0))), *'(s(0), s(0))) by F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), 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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, -'_2, +'_2, F_1 Compound Symbols: c5_2, c1_1, c3_1, c7_3, c8_3 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) by -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, +'_2, F_1, -'_2 Compound Symbols: c5_2, c3_1, c7_3, c8_3, c1_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) by +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c7_3, c8_3, c1_1, c3_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), *(s(s(z0)), s(z0)))), *'(s(s(z0)), s(s(z0)))) by F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c8(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c8(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c8(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c8(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c8(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(s(z0))) -> c8(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c7(F(-(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0))))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(s(z0))) -> c7(F(-(p(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c7(F(-(p(*(s(0), s(0))), +(s(0), 0))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) F(s(z0)) -> c7(F(-(p(+(s(z0), *(s(z0), z0))), +(s(z0), *(s(z0), z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(z0)) -> c7(F(-(p(s(+(z0, *(s(z0), z0)))), *(s(z0), s(z0)))), -'(p(*(s(z0), s(z0))), s(+(z0, *(s(z0), z0)))), *'(s(z0), s(z0))) F(s(0)) -> c7(F(-(p(+(s(0), 0)), *(s(0), s(0)))), -'(p(*(s(0), s(0))), s(+(0, *(s(0), 0)))), *'(s(0), s(0))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) F(s(s(z0))) -> c7(F(-(p(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), *(s(s(z0)), s(s(z0))))), -'(p(*(s(s(z0)), s(s(z0)))), s(+(s(z0), *(s(s(z0)), s(z0))))), *'(s(s(z0)), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, p_1 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c8_3, c7_3, c1_1, c3_1