KILLED proof of input_uaq8MENqci.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 37 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 344 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 266 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 30.1 s] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 4212 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 149 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 39 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 72 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (82) CdtProblem (83) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 87 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 63 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (170) 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: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) fibo(0) -> fib(0) fibo(s(0)) -> fib(s(0)) fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) fib(0) -> s(0) fib(s(0)) -> s(0) fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) sum(x, 0) -> x sum(x, s(y)) -> s(sum(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: lt(0', s(x)) -> true lt(x, 0') -> false lt(s(x), s(y)) -> lt(x, y) fibo(0') -> fib(0') fibo(s(0')) -> fib(s(0')) fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) fib(0') -> s(0') fib(s(0')) -> s(0') fib(s(s(x))) -> if(true, 0', s(s(x)), 0', 0') if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) sum(x, 0') -> x sum(x, s(y)) -> s(sum(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: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) fibo(0) -> fib(0) fibo(s(0)) -> fib(s(0)) fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) fib(0) -> s(0) fib(s(0)) -> s(0) fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) sum(x, 0) -> x sum(x, s(y)) -> s(sum(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: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fibo(0) -> fib(0) [1] fibo(s(0)) -> fib(s(0)) [1] fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1] fib(0) -> s(0) [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1] if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1] if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1] sum(x, 0) -> x [1] sum(x, s(y)) -> s(sum(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fibo(0) -> fib(0) [1] fibo(s(0)) -> fib(s(0)) [1] fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1] fib(0) -> s(0) [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1] if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1] if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1] sum(x, 0) -> x [1] sum(x, s(y)) -> s(sum(x, y)) [1] The TRS has the following type information: lt :: 0:s -> 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s true :: true:false false :: true:false fibo :: 0:s -> 0:s fib :: 0:s -> 0:s sum :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: fibo_1 lt_2 fib_1 if_5 sum_2 Due to the following rules being added: if(v0, v1, v2, v3, v4) -> 0 [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fibo(0) -> fib(0) [1] fibo(s(0)) -> fib(s(0)) [1] fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1] fib(0) -> s(0) [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1] if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1] if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1] sum(x, 0) -> x [1] sum(x, s(y)) -> s(sum(x, y)) [1] if(v0, v1, v2, v3, v4) -> 0 [0] The TRS has the following type information: lt :: 0:s -> 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s true :: true:false false :: true:false fibo :: 0:s -> 0:s fib :: 0:s -> 0:s sum :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fibo(0) -> fib(0) [1] fibo(s(0)) -> fib(s(0)) [1] fibo(s(s(0))) -> sum(fib(s(0)), fib(0)) [3] fibo(s(s(s(0)))) -> sum(sum(fibo(s(0)), fibo(0)), fib(s(0))) [3] fibo(s(s(s(s(x''))))) -> sum(sum(fibo(s(s(x''))), fibo(s(x''))), sum(fibo(s(x'')), fibo(x''))) [3] fib(0) -> s(0) [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1] if(true, c, s(s(x)), a, b) -> if(lt(c, s(x)), s(c), s(s(x)), b, c) [2] if(false, c, s(s(x)), 0, 0) -> sum(fib(0), fib(0)) [3] if(false, c, s(s(x)), 0, s(0)) -> sum(fib(0), fib(s(0))) [3] if(false, c, s(s(x)), 0, s(s(x2))) -> sum(fib(0), sum(fibo(s(x2)), fibo(x2))) [3] if(false, c, s(s(x)), s(0), 0) -> sum(fib(s(0)), fib(0)) [3] if(false, c, s(s(x)), s(0), s(0)) -> sum(fib(s(0)), fib(s(0))) [3] if(false, c, s(s(x)), s(0), s(s(x3))) -> sum(fib(s(0)), sum(fibo(s(x3)), fibo(x3))) [3] if(false, c, s(s(x)), s(s(x1)), 0) -> sum(sum(fibo(s(x1)), fibo(x1)), fib(0)) [3] if(false, c, s(s(x)), s(s(x1)), s(0)) -> sum(sum(fibo(s(x1)), fibo(x1)), fib(s(0))) [3] if(false, c, s(s(x)), s(s(x1)), s(s(x4))) -> sum(sum(fibo(s(x1)), fibo(x1)), sum(fibo(s(x4)), fibo(x4))) [3] sum(x, 0) -> x [1] sum(x, s(y)) -> s(sum(x, y)) [1] if(v0, v1, v2, v3, v4) -> 0 [0] The TRS has the following type information: lt :: 0:s -> 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s true :: true:false false :: true:false fibo :: 0:s -> 0:s fib :: 0:s -> 0:s sum :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + x), 0, 0) :|: x >= 0, z = 1 + (1 + x) fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + x'')), fibo(1 + x'')), sum(fibo(1 + x''), fibo(x''))) :|: z = 1 + (1 + (1 + (1 + x''))), x'' >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), sum(fibo(1 + x4), fibo(x4))) :|: x1 >= 0, x4 >= 0, z2 = 1 + (1 + x4), c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1) if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), fib(0)) :|: x1 >= 0, c >= 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1) if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), fib(1 + 0)) :|: z2 = 1 + 0, x1 >= 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1) if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + x2), fibo(x2))) :|: z1 = 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z2 = 1 + (1 + x2), x2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, c >= 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + x3), fibo(x3))) :|: z2 = 1 + (1 + x3), c >= 0, z1 = 1 + 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, x3 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: c >= 0, z1 = 1 + 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, c >= 0, z1 = 1 + 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(c, 1 + x), 1 + c, 1 + (1 + x), b, c) :|: b >= 0, z2 = b, c >= 0, a >= 0, z = 1, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 sum(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt } { sum } { fib, if, fibo } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo} Previous analysis results are: lt: runtime: ?, size: O(1) [1] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {sum}, {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {sum}, {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {sum}, {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] sum: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] sum: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] sum: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (33) 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: ? Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: fibo after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] sum: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] fib: runtime: ?, size: EXP if: runtime: ?, size: INF fibo: runtime: ?, size: INF ---------------------------------------- (35) 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: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0)) fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0 fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0 if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {fib,if,fibo} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] sum: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] fib: runtime: INF, size: EXP if: runtime: ?, size: INF fibo: runtime: ?, size: INF ---------------------------------------- (37) 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: if(v0, v1, v2, v3, v4) -> null_if [0] lt(v0, v1) -> null_lt [0] fibo(v0) -> null_fibo [0] fib(v0) -> null_fib [0] sum(v0, v1) -> null_sum [0] And the following fresh constants: null_if, null_lt, null_fibo, null_fib, null_sum ---------------------------------------- (38) 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: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fibo(0) -> fib(0) [1] fibo(s(0)) -> fib(s(0)) [1] fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1] fib(0) -> s(0) [1] fib(s(0)) -> s(0) [1] fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1] if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1] if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1] sum(x, 0) -> x [1] sum(x, s(y)) -> s(sum(x, y)) [1] if(v0, v1, v2, v3, v4) -> null_if [0] lt(v0, v1) -> null_lt [0] fibo(v0) -> null_fibo [0] fib(v0) -> null_fib [0] sum(v0, v1) -> null_sum [0] The TRS has the following type information: lt :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> true:false:null_lt 0 :: 0:s:null_if:null_fibo:null_fib:null_sum s :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum true :: true:false:null_lt false :: true:false:null_lt fibo :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum fib :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum sum :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum if :: true:false:null_lt -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum null_if :: 0:s:null_if:null_fibo:null_fib:null_sum null_lt :: true:false:null_lt null_fibo :: 0:s:null_if:null_fibo:null_fib:null_sum null_fib :: 0:s:null_if:null_fibo:null_fib:null_sum null_sum :: 0:s:null_if:null_fibo:null_fib:null_sum Rewrite Strategy: INNERMOST ---------------------------------------- (39) 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 true => 2 false => 1 null_if => 0 null_lt => 0 null_fibo => 0 null_fib => 0 null_sum => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> if(2, 0, 1 + (1 + x), 0, 0) :|: x >= 0, z = 1 + (1 + x) fib(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 fib(z) -{ 1 }-> 1 + 0 :|: z = 0 fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 fibo(z) -{ 1 }-> sum(fibo(1 + x), fibo(x)) :|: x >= 0, z = 1 + (1 + x) fibo(z) -{ 1 }-> fib(0) :|: z = 0 fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0 fibo(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 if(z, z', z'', z1, z2) -{ 1 }-> sum(fibo(a), fibo(b)) :|: b >= 0, z2 = b, c >= 0, a >= 0, z = 1, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a if(z, z', z'', z1, z2) -{ 1 }-> if(lt(1 + c, 1 + (1 + x)), 1 + c, 1 + (1 + x), b, c) :|: z = 2, b >= 0, z2 = b, c >= 0, a >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 sum(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 sum(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 sum(z, z') -{ 1 }-> 1 + sum(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) fibo(0) -> fib(0) fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fib(0) -> s(0) fib(s(0)) -> s(0) fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0) if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0) if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3)) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) Tuples: LT(0, s(z0)) -> c LT(z0, 0) -> c1 LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(0) -> c3(FIB(0)) FIBO(s(0)) -> c4(FIB(s(0))) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) FIB(0) -> c7 FIB(s(0)) -> c8 FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3)) SUM(z0, 0) -> c13 SUM(z0, s(z1)) -> c14(SUM(z0, z1)) S tuples: LT(0, s(z0)) -> c LT(z0, 0) -> c1 LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(0) -> c3(FIB(0)) FIBO(s(0)) -> c4(FIB(s(0))) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) FIB(0) -> c7 FIB(s(0)) -> c8 FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3)) SUM(z0, 0) -> c13 SUM(z0, s(z1)) -> c14(SUM(z0, z1)) K tuples:none Defined Rule Symbols: lt_2, fibo_1, fib_1, if_5, sum_2 Defined Pair Symbols: LT_2, FIBO_1, FIB_1, IF_5, SUM_2 Compound Symbols: c, c1, c2_1, c3_1, c4_1, c5_2, c6_2, c7, c8, c9_1, c10_2, c11_2, c12_2, c13, c14_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0)) Removed 7 trailing nodes: FIB(s(0)) -> c8 FIB(0) -> c7 LT(z0, 0) -> c1 FIBO(0) -> c3(FIB(0)) LT(0, s(z0)) -> c FIBO(s(0)) -> c4(FIB(s(0))) SUM(z0, 0) -> c13 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) fibo(0) -> fib(0) fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fib(0) -> s(0) fib(s(0)) -> s(0) fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0) if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0) if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3)) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) K tuples:none Defined Rule Symbols: lt_2, fibo_1, fib_1, if_5, sum_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c5_2, c6_2, c10_2, c11_2, c12_2, c14_1 ---------------------------------------- (45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) fibo(0) -> fib(0) fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fib(0) -> s(0) fib(s(0)) -> s(0) fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0) if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0) if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3)) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) K tuples:none Defined Rule Symbols: lt_2, fibo_1, fib_1, if_5, sum_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c5_2, c6_2, c10_2, c14_1, c_1 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0) if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0) if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) K tuples:none Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c5_2, c6_2, c10_2, c14_1, c_1 ---------------------------------------- (49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [3] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [3] + x_1 + [3]x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(fib(x_1)) = [3] POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(sum(x_1, x_2)) = [3] POL(true) = [1] ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c5_2, c6_2, c10_2, c14_1, c_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) by FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c6_2, c10_2, c14_1, c_1, c5_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c6_2, c10_2, c14_1, c_1, c5_2, c5_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(sum(x_1, x_2)) = 0 POL(true) = [1] ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, FIBO_1, IF_5, SUM_2 Compound Symbols: c2_1, c6_2, c10_2, c14_1, c_1, c5_2, c5_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) by FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0)) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0)) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0)) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, IF_5, SUM_2, FIBO_1 Compound Symbols: c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, IF_5, SUM_2, FIBO_1 Compound Symbols: c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1 ---------------------------------------- (61) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, IF_5, SUM_2, FIBO_1 Compound Symbols: c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(sum(x_1, x_2)) = [1] POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, IF_5, SUM_2, FIBO_1 Compound Symbols: c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) by IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) by IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) by FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2 ---------------------------------------- (73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = [1] POL(s(x_1)) = [1] POL(sum(x_1, x_2)) = 0 POL(true) = [1] ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) by FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) by FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1 ---------------------------------------- (83) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c4_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = 0 POL(s(x_1)) = [1] POL(sum(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c4_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) by FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0))) FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1 ---------------------------------------- (91) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1 ---------------------------------------- (93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) We considered the (Usable) Rules:none And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(false) = [1] POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = 0 POL(sum(x_1, x_2)) = 0 POL(true) = [1] ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) by FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) by FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) by FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) by FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) by IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (105) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) We considered the (Usable) Rules: lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) lt(z0, 0) -> false And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBO(x_1)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 POL(LT(x_1, x_2)) = 0 POL(SUM(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(false) = 0 POL(fib(x_1)) = 0 POL(fibo(x_1)) = 0 POL(lt(x_1, x_2)) = 0 POL(s(x_1)) = [1] POL(sum(x_1, x_2)) = 0 POL(true) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) by IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) by IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) by IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) by IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(0), s(s(z0))) -> c(SUM(fib(s(0)), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) by IF(false, x0, s(s(x1)), s(s(x2)), s(0)) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(x2)), s(s(z0))) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), s(s(x2)), 0) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), fib(0))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) by IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(s(z0))) -> c(SUM(fib(0), sum(fibo(s(z0)), fibo(z0)))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) by FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) by FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (127) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) by IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, IF_5, FIBO_1 Compound Symbols: c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1 ---------------------------------------- (129) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) by IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c5_2, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c5_2, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) by FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) by FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: LT_2, SUM_2, FIBO_1, IF_5 Compound Symbols: c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(z0), s(z1)) -> c2(LT(z0, z1)) by LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) S tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: SUM_2, FIBO_1, IF_5, LT_2 Compound Symbols: c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) by FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) S tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: SUM_2, FIBO_1, IF_5, LT_2 Compound Symbols: c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) by FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0)))) S tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: SUM_2, FIBO_1, IF_5, LT_2 Compound Symbols: c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) by FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0)))) S tuples: SUM(z0, s(z1)) -> c14(SUM(z0, z1)) FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: SUM_2, FIBO_1, IF_5, LT_2 Compound Symbols: c14_1, c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SUM(z0, s(z1)) -> c14(SUM(z0, z1)) by SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0)))) SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1))) S tuples: FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1))) K tuples: FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: FIBO_1, IF_5, LT_2, SUM_2 Compound Symbols: c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1, c14_1 ---------------------------------------- (169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0))) IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0))) FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0))) FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0))) FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: fibo(s(0)) -> fib(s(0)) fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0)) fibo(0) -> fib(0) fib(s(0)) -> s(0) fib(0) -> s(0) sum(z0, 0) -> z0 sum(z0, s(z1)) -> s(sum(z0, z1)) lt(s(z0), s(z1)) -> lt(z0, z1) lt(0, s(z0)) -> true lt(z0, 0) -> false Tuples: FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0))) FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0))) FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0)))) IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0)))) IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0))))) IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0)))) IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0)))) IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))))) IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3)))) IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0))) IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0))) IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0))) IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2))) IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3))) IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3))) IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(0))) -> c5(SUM(s(0), fib(0))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0)))) FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0)))) SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1))) S tuples: FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0)))) FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0))))) FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0)))) FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0)))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0)))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0))))) FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0)))) IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1)))) FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0)))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1))) K tuples: FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0))))) FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0))))) IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1)))) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3)) IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0))) Defined Rule Symbols: fibo_1, fib_1, sum_2, lt_2 Defined Pair Symbols: FIBO_1, IF_5, LT_2, SUM_2 Compound Symbols: c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1, c14_1