KILLED proof of input_vN0YNqW3J4.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 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), 261 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 51 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 1599 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 290 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 101 ms] (42) CdtProblem (43) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 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) CdtRewritingProof [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), 1 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(x))) -> +(fib(s(x)), fib(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fib(0') -> 0' fib(s(0')) -> s(0') fib(s(s(x))) -> +'(fib(s(x)), fib(x)) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(x))) -> +(fib(s(x)), fib(x)) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> +(fib(s(x)), fib(x)) [1] +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (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: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> plus(fib(s(x)), fib(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> plus(fib(s(x)), fib(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: fib :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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: none (c) The following functions are completely defined: fib_1 plus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> plus(fib(s(x)), fib(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: fib :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(0))) -> plus(s(0), 0) [3] fib(s(s(s(0)))) -> plus(plus(fib(s(0)), fib(0)), s(0)) [3] fib(s(s(s(s(x''))))) -> plus(plus(fib(s(s(x''))), fib(s(x''))), plus(fib(s(x'')), fib(x''))) [3] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: fib :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s 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 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + x'')), fib(1 + x'')), plus(fib(1 + x''), fib(x''))) :|: z = 1 + (1 + (1 + (1 + x''))), x'' >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 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: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { fib } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {fib} ---------------------------------------- (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: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {fib} ---------------------------------------- (23) 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' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {fib} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (25) 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' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 3 }-> plus(1 + 0, 0) :|: z = 1 + (1 + 0) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 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: fib(z) -{ 4 }-> s :|: s >= 0, s <= 1 + 0 + 0, z = 1 + (1 + 0) fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: fib after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 4 }-> s :|: s >= 0, s <= 1 + 0 + 0, z = 1 + (1 + 0) fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] fib: runtime: ?, size: EXP ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fib after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 4 }-> s :|: s >= 0, s <= 1 + 0 + 0, z = 1 + (1 + 0) fib(z) -{ 3 }-> plus(plus(fib(1 + 0), fib(0)), 1 + 0) :|: z = 1 + (1 + (1 + 0)) fib(z) -{ 3 }-> plus(plus(fib(1 + (1 + (z - 4))), fib(1 + (z - 4))), plus(fib(1 + (z - 4)), fib(z - 4))) :|: z - 4 >= 0 fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] fib: runtime: INF, size: EXP ---------------------------------------- (33) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (34) 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: fib(0) -> 0 [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> plus(fib(s(x)), fib(x)) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: fib :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> plus(fib(1 + x), fib(x)) :|: x >= 0, z = 1 + (1 + x) fib(z) -{ 1 }-> 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(0) -> c FIB(s(0)) -> c1 FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, 0) -> c4 +'(z0, s(z1)) -> c5(+'(z0, z1)) S tuples: FIB(0) -> c FIB(s(0)) -> c1 FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, 0) -> c4 +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c, c1, c2_2, c3_2, c4, c5_1 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: FIB(s(0)) -> c1 FIB(0) -> c +'(z0, 0) -> c4 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) S tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) We considered the (Usable) Rules:none And the Tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = [1] POL(0) = [1] POL(FIB(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(fib(x_1)) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) S tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1 ---------------------------------------- (43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) We considered the (Usable) Rules:none And the Tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(0) = [1] POL(FIB(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(fib(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) by FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(0))) -> c2(+'(fib(s(0)), 0), FIB(s(0))) FIB(s(s(0))) -> c2(+'(s(0), fib(0)), FIB(s(0))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(0))) -> c2(+'(fib(s(0)), 0), FIB(s(0))) FIB(s(s(0))) -> c2(+'(s(0), fib(0)), FIB(s(0))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0))) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c3_2, c5_1, c2_2 ---------------------------------------- (47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(0))) -> c2(+'(fib(s(0)), 0), FIB(s(0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(0))) -> c2(+'(s(0), fib(0)), FIB(s(0))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c3_2, c5_1, c2_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c3_2, c5_1, c2_2, c2_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) by FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0)), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c3(+'(fib(s(0)), 0), FIB(0)) FIB(s(s(0))) -> c3(+'(s(0), fib(0)), FIB(0)) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0)), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c3(+'(fib(s(0)), 0), FIB(0)) FIB(s(s(0))) -> c3(+'(s(0), fib(0)), FIB(0)) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples: FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0)) Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c3_2 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(0))) -> c3(+'(fib(s(0)), 0), FIB(0)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0)), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c3(+'(s(0), fib(0)), FIB(0)) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c3_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c2(+'(fib(s(s(0))), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c3_2, c3_1 ---------------------------------------- (57) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c3_2, c3_1, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(s(z0))))) -> c2(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c3_2, c3_1, c_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(z0)))) -> c2(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(s(z0)))) by FIB(s(s(s(0)))) -> c2(+'(+(fib(s(0)), fib(0)), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c2(+'(+(fib(s(0)), 0), fib(s(0))), FIB(s(s(0)))) FIB(s(s(s(0)))) -> c2(+'(+(s(0), fib(0)), fib(s(0))), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c2(+'(+(fib(s(0)), fib(0)), s(0)), FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c2(+'(+(fib(s(0)), 0), fib(s(0))), FIB(s(s(0)))) FIB(s(s(s(0)))) -> c2(+'(+(s(0), fib(0)), fib(s(0))), FIB(s(s(0)))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c3_1, c_1, c2_2 ---------------------------------------- (63) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(0))) -> c2(+'(s(0), fib(0))) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c3_1, c_1, c2_2, c1_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(0))) -> c2(+'(s(0), fib(0))) by FIB(s(s(0))) -> c2(+'(s(0), 0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(0))) -> c2(+'(s(0), 0)) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c2_2, c2_1, c1_1 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(0))) -> c2(+'(s(0), 0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c2_2, c2_1, c1_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(s(z0))))) -> c3(+'(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) by FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(0)))))) -> c3(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c3(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c3(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(0)))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(0)))))) -> c3(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c3(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c3(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(0)))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c2_2, c2_1, c1_1 ---------------------------------------- (71) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(0)))))) -> c3(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c2_2, c2_1, c1_1, c4_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(z0)))) -> c3(+'(+(fib(s(z0)), fib(z0)), fib(s(z0))), FIB(s(z0))) by FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0)), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(s(0))))) -> c3(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0))), FIB(s(0))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0))), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(0)))))) -> c3(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0)), FIB(s(0))) FIB(s(s(s(s(0))))) -> c3(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0))), FIB(s(0))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0))), FIB(s(0))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(fib(s(0)), 0)), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(0))))) -> c2(+'(fib(s(s(s(0)))), +(s(0), fib(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c2(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(0)))))) -> c3(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c3(+'(+(fib(s(s(0))), s(0)), fib(s(s(0)))), FIB(s(s(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1 ---------------------------------------- (77) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1, c6_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(0)))) -> c3(+'(fib(s(s(0))), s(0))) by FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(0))) -> c3(+'(s(0), fib(0))) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c3_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1, c6_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(0))) -> c3(+'(s(0), fib(0))) by FIB(s(s(0))) -> c3(+'(s(0), 0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(0)))) FIB(s(s(0))) -> c3(+'(s(0), 0)) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: FIB(s(s(s(0)))) -> c(FIB(s(s(0)))) FIB(s(s(0))) -> c3(+'(s(0), 0)) FIB(s(s(s(0)))) -> c1(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c4(FIB(s(s(0)))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(s(s(s(0)))) -> c(+'(fib(s(s(0))), s(0))) by FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) by FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0))), FIB(s(s(s(s(0)))))) by FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c2(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) by FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_2, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(z0))))) -> c2(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c2_2, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c2_2, c1_1, c3_2, c4_1, c3_1, c6_1, c_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), fib(0)), s(0))) by FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), fib(s(0)))) by FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c1_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), fib(s(0)))) by FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2, c1_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2, c1_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c3(+'(fib(s(s(s(s(z0))))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2, c1_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) by FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c5(+'(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: +'_2, FIB_1 Compound Symbols: c5_1, c2_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2, c1_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(z1)) -> c5(+'(z0, z1)) by +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(x0))))) -> c3(+'(+(fib(s(s(x0))), fib(s(x0))), +(fib(s(x0)), fib(x0))), FIB(s(s(x0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_2, c4_1, c3_1, c6_1, c_1, c2_2, c1_1, c5_1 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: FIB(s(s(s(0)))) -> c1(+'(+(fib(s(0)), 0), s(0))) FIB(s(s(s(0)))) -> c1(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), fib(0)), s(0))) FIB(s(s(s(0)))) -> c(+'(+(fib(s(0)), fib(0)), s(0))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(z0))))) -> c3(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c4_1, c3_2, c3_1, c6_1, c2_2, c5_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) by FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(z0))))) -> c3(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c4_1, c3_2, c3_1, c6_1, c2_2, c5_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) by FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_2, c3_1, c6_1, c2_2, c5_1, c4_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(z0))))) -> c3(+'(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) by FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_2, c3_1, c6_1, c2_2, c5_1, c4_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), fib(s(s(s(z0))))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_2, c3_1, c6_1, c2_2, c5_1, c4_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), fib(s(s(z0)))), FIB(s(s(z0)))) by FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), fib(s(0)))) by FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), s(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), s(0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(s(0)))) -> c3(+'(+(fib(s(0)), 0), s(0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c3_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), fib(s(0)))) by FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), s(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), s(0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1, c3_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: FIB(s(s(s(s(0))))) -> c6(FIB(s(s(s(0))))) FIB(s(s(s(0)))) -> c3(+'(+(s(0), fib(0)), s(0))) FIB(s(s(s(s(s(0)))))) -> c6(FIB(s(s(s(0))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(fib(s(0)), 0))) by FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), fib(0)))) by FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) by FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), +(fib(s(s(0))), s(0)))) by FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c6_1, c2_2, c3_2, c5_1, c4_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), fib(s(s(0))))) by FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) by FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0))), FIB(s(s(s(s(0)))))) by FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(fib(s(s(z0))), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) by FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), fib(s(s(z0)))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), fib(s(0)))) by FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c4_1, c6_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), +(s(0), 0))) by FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c6_1, c4_1 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(s(s(0))))) -> c4(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c6_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(fib(s(s(z0))), fib(s(z0)))), FIB(s(s(s(z0))))) by FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_1, c2_2, c3_2, c5_1, c6_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIB(s(s(s(s(x0))))) -> c2(FIB(s(s(s(x0))))) by FIB(s(s(s(s(s(y0)))))) -> c2(FIB(s(s(s(s(y0)))))) FIB(s(s(s(s(s(0)))))) -> c2(FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(s(0))))))) -> c2(FIB(s(s(s(s(s(0))))))) FIB(s(s(s(s(s(s(s(y0)))))))) -> c2(FIB(s(s(s(s(s(s(y0)))))))) FIB(s(s(s(s(s(s(y0))))))) -> c2(FIB(s(s(s(s(s(y0))))))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(y0)))))) -> c2(FIB(s(s(s(s(y0)))))) FIB(s(s(s(s(s(0)))))) -> c2(FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(s(0))))))) -> c2(FIB(s(s(s(s(s(0))))))) FIB(s(s(s(s(s(s(s(y0)))))))) -> c2(FIB(s(s(s(s(s(s(y0)))))))) FIB(s(s(s(s(s(s(y0))))))) -> c2(FIB(s(s(s(s(s(y0))))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c2_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), fib(s(0)))) by FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(y0)))))) -> c2(FIB(s(s(s(s(y0)))))) FIB(s(s(s(s(s(0)))))) -> c2(FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(s(0))))))) -> c2(FIB(s(s(s(s(s(0))))))) FIB(s(s(s(s(s(s(s(y0)))))))) -> c2(FIB(s(s(s(s(s(s(y0)))))))) FIB(s(s(s(s(s(s(y0))))))) -> c2(FIB(s(s(s(s(s(y0))))))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), s(0))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c2_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), s(0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: fib(0) -> 0 fib(s(0)) -> s(0) fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FIB(s(s(s(s(z0))))) -> c2(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(s(z0))))) FIB(s(s(s(s(z0))))) -> c3(+'(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0))), FIB(s(s(z0)))) +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) FIB(s(s(s(s(0))))) -> c6(+'(fib(s(s(s(0)))), +(s(0), 0))) FIB(s(s(s(s(0))))) -> c6(+'(+(fib(s(s(0))), s(0)), +(fib(s(0)), fib(0)))) FIB(s(s(s(s(s(0)))))) -> c6(+'(fib(s(s(s(s(0))))), s(+(fib(s(s(0))), 0)))) FIB(s(s(s(s(s(s(z0))))))) -> c2(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(s(z0))))))) FIB(s(s(s(s(s(0)))))) -> c2(+'(fib(s(s(s(s(0))))), s(fib(s(s(0))))), FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(z0)))))) -> c2(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(s(z0))))))) -> c3(+'(fib(s(s(s(s(s(z0)))))), +(+(+(fib(s(z0)), fib(z0)), fib(s(z0))), +(fib(s(z0)), fib(z0)))), FIB(s(s(s(s(z0)))))) FIB(s(s(s(s(s(z0)))))) -> c3(+'(+(fib(s(s(s(z0)))), +(fib(s(z0)), fib(z0))), +(+(fib(s(z0)), fib(z0)), fib(s(z0)))), FIB(s(s(s(z0))))) FIB(s(s(s(s(s(y0)))))) -> c2(FIB(s(s(s(s(y0)))))) FIB(s(s(s(s(s(0)))))) -> c2(FIB(s(s(s(s(0)))))) FIB(s(s(s(s(s(s(0))))))) -> c2(FIB(s(s(s(s(s(0))))))) FIB(s(s(s(s(s(s(s(y0)))))))) -> c2(FIB(s(s(s(s(s(s(y0)))))))) FIB(s(s(s(s(s(s(y0))))))) -> c2(FIB(s(s(s(s(s(y0))))))) S tuples: +'(z0, s(s(y1))) -> c5(+'(z0, s(y1))) K tuples:none Defined Rule Symbols: fib_1, +_2 Defined Pair Symbols: FIB_1, +'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c2_1