KILLED proof of input_D9e6sOjorZ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 69 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 190 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 16 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 315 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 37 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 518 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 2661 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 1476 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 155 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms] (86) CdtProblem (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRewritingProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) 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: +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +(0, y) -> y [1] +(s(x), y) -> s(+(x, y)) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(x, *(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus * => times - => minus ---------------------------------------- (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: 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] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [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: 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] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 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: twice_1 f_1 (c) The following functions are completely defined: times_2 minus_2 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 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: 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] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(s(x)) -> f(minus(plus(s(s(x)), times(s(s(x)), s(x))), plus(plus(s(x), times(s(x), s(x))), s(s(0))))) [3] minus(v0, v1) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 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(plus(1 + (1 + x), times(1 + (1 + x), 1 + x)), plus(plus(1 + x, times(1 + x, 1 + x)), 1 + (1 + 0)))) :|: 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 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { twice } { minus } { plus } { times } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {plus}, {times}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {plus}, {times}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {plus}, {times}, {f} Previous analysis results are: twice: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) 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' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (35) 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' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (37) 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 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] 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'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] 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'] ---------------------------------------- (41) 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' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (43) 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' ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= 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 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 }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (47) 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 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= 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 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 }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (49) 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: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= 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 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 }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: INF, size: O(1) [0] ---------------------------------------- (51) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] twice(v0) -> null_twice [0] And the following fresh constants: null_minus, null_f, null_plus, null_times, null_twice ---------------------------------------- (52) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(0, 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] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [1] minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] twice(v0) -> null_twice [0] The TRS has the following type information: plus :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice 0 :: 0:s:null_minus:null_plus:null_times:null_twice s :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice times :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice twice :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice minus :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice f :: 0:s:null_minus:null_plus:null_times:null_twice -> null_f null_minus :: 0:s:null_minus:null_plus:null_times:null_twice null_f :: null_f null_plus :: 0:s:null_minus:null_plus:null_times:null_twice null_times :: 0:s:null_minus:null_plus:null_times:null_twice null_twice :: 0:s:null_minus:null_plus:null_times:null_twice Rewrite Strategy: INNERMOST ---------------------------------------- (53) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_minus => 0 null_f => 0 null_plus => 0 null_times => 0 null_twice => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(minus(times(1 + (1 + x), 1 + (1 + x)), plus(times(1 + x, 1 + (1 + x)), 1 + (1 + 0)))) :|: 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 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 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))) Tuples: +'(0, z0) -> c +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, 0) -> c2 *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(0) -> c4 TWICE(s(z0)) -> c5(TWICE(z0)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: +'(0, z0) -> c +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, 0) -> c2 *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(0) -> c4 TWICE(s(z0)) -> c5(TWICE(z0)) -'(z0, 0) -> c6 -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples:none Defined Rule Symbols: +_2, *_2, twice_1, -_2, f_1 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c, c1_1, c2, c3_2, c4, c5_1, c6, c7_1, c8_3, c9_4 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: *'(z0, 0) -> c2 -'(z0, 0) -> c6 TWICE(0) -> c4 +'(0, z0) -> c ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples:none Defined Rule Symbols: +_2, *_2, twice_1, -_2, f_1 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TWICE(s(z0)) -> c5(TWICE(z0)) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = 0 POL(TWICE(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4, c8_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(s(x_1)) = 0 ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c9_4, c8_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c8_2 ---------------------------------------- (73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(s(x_1)) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c9_3 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c2_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(s(x_1)) = 0 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c2_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(s(x_1)) = 0 ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) ---------------------------------------- (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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(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) Tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) S tuples: +'(s(z0), z1) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, TWICE_1, -'_2, F_1 Compound Symbols: c1_1, c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(z0), z1) -> c1(+'(z0, z1)) by +'(s(s(y0)), z1) -> c1(+'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1, c1_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1, c1_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) by F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) TWICE(s(z0)) -> c5(TWICE(z0)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: TWICE(s(z0)) -> c5(TWICE(z0)) F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, TWICE_1, -'_2, F_1, +'_2 Compound Symbols: c3_2, c5_1, c7_1, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TWICE(s(z0)) -> c5(TWICE(z0)) by TWICE(s(s(y0))) -> c5(TWICE(s(y0))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(z0), s(z1)) -> c7(-'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, -'_2, F_1, +'_2, TWICE_1 Compound Symbols: c3_2, c7_1, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c7(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1, c7_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1, c7_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c9_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c1_1, c9_4, c5_1, c7_1, c9_1, c8_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c1_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(s(y0)), z1) -> c1(+'(s(y0), z1)) by +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1, c1_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1, c1_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(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) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) by F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(y0))) -> c5(TWICE(s(y0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, TWICE_1, -'_2, +'_2 Compound Symbols: c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TWICE(s(s(y0))) -> c5(TWICE(s(y0))) by TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2, TWICE_1 Compound Symbols: c3_2, c_1, c2_1, c9_4, c7_1, c9_1, c8_3, c8_1, c1_1, c5_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2, TWICE_1 Compound Symbols: c3_2, c_1, c2_1, c7_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2, TWICE_1 Compound Symbols: c3_2, c_1, c2_1, c7_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2, TWICE_1 Compound Symbols: c3_2, c_1, c2_1, c7_1, c9_1, c8_3, c8_1, c1_1, c9_4, c5_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2, TWICE_1 Compound Symbols: c3_2, c_1, c2_1, c7_1, c8_3, c8_1, c1_1, c9_4, c5_1, c9_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) by -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) Tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0)))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))) -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1)) F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1)))) K tuples: F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c(*'(s(s(x0)), s(s(x0)))) F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) F(s(x0)) -> c2(*'(s(x0), s(s(x0)))) F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0)))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, TWICE_1, -'_2 Compound Symbols: c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c9_1, c7_1