KILLED proof of input_L8ZEZcidUl.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 2 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 198 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 932 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 17 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 386 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (62) CdtProblem (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 314 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2083 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (94) CdtProblem (95) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1071 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (174) CdtProblem (175) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2087 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2689 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 37 ms] (192) CdtProblem (193) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (230) 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: tower(x) -> f(a, x, s(0)) f(a, 0, y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0) -> s(0) exp(s(x)) -> double(exp(x)) double(0) -> 0 double(s(x)) -> s(s(double(x))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(x))) -> s(half(x)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0)) f(a, 0, y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0) -> s(0) exp(s(x)) -> double(exp(x)) double(0) -> 0 double(s(x)) -> s(s(double(x))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(x))) -> s(half(x)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0)) [1] f(a, 0, y) -> y [1] f(a, s(x), y) -> f(b, y, s(x)) [1] f(b, y, x) -> f(a, half(x), exp(y)) [1] exp(0) -> s(0) [1] exp(s(x)) -> double(exp(x)) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] half(0) -> double(0) [1] half(s(0)) -> half(0) [1] half(s(s(x))) -> s(half(x)) [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: tower(x) -> f(a, x, s(0)) [1] f(a, 0, y) -> y [1] f(a, s(x), y) -> f(b, y, s(x)) [1] f(b, y, x) -> f(a, half(x), exp(y)) [1] exp(0) -> s(0) [1] exp(s(x)) -> double(exp(x)) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] half(0) -> double(0) [1] half(s(0)) -> half(0) [1] half(s(s(x))) -> s(half(x)) [1] The TRS has the following type information: tower :: 0:s -> 0:s f :: a:b -> 0:s -> 0:s -> 0:s a :: a:b s :: 0:s -> 0:s 0 :: 0:s b :: a:b half :: 0:s -> 0:s exp :: 0:s -> 0:s double :: 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: tower_1 f_3 (c) The following functions are completely defined: exp_1 half_1 double_1 Due to the following rules being added: none 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: tower(x) -> f(a, x, s(0)) [1] f(a, 0, y) -> y [1] f(a, s(x), y) -> f(b, y, s(x)) [1] f(b, y, x) -> f(a, half(x), exp(y)) [1] exp(0) -> s(0) [1] exp(s(x)) -> double(exp(x)) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] half(0) -> double(0) [1] half(s(0)) -> half(0) [1] half(s(s(x))) -> s(half(x)) [1] The TRS has the following type information: tower :: 0:s -> 0:s f :: a:b -> 0:s -> 0:s -> 0:s a :: a:b s :: 0:s -> 0:s 0 :: 0:s b :: a:b half :: 0:s -> 0:s exp :: 0:s -> 0:s double :: 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: tower(x) -> f(a, x, s(0)) [1] f(a, 0, y) -> y [1] f(a, s(x), y) -> f(b, y, s(x)) [1] f(b, 0, 0) -> f(a, double(0), s(0)) [3] f(b, s(x''), 0) -> f(a, double(0), double(exp(x''))) [3] f(b, 0, s(0)) -> f(a, half(0), s(0)) [3] f(b, s(x1), s(0)) -> f(a, half(0), double(exp(x1))) [3] f(b, 0, s(s(x'))) -> f(a, s(half(x')), s(0)) [3] f(b, s(x2), s(s(x'))) -> f(a, s(half(x')), double(exp(x2))) [3] exp(0) -> s(0) [1] exp(s(0)) -> double(s(0)) [2] exp(s(s(x3))) -> double(double(exp(x3))) [2] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] half(0) -> double(0) [1] half(s(0)) -> half(0) [1] half(s(s(x))) -> s(half(x)) [1] The TRS has the following type information: tower :: 0:s -> 0:s f :: a:b -> 0:s -> 0:s -> 0:s a :: a:b s :: 0:s -> 0:s 0 :: 0:s b :: a:b half :: 0:s -> 0:s exp :: 0:s -> 0:s double :: 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: a => 0 0 => 0 b => 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x exp(z) -{ 2 }-> double(double(exp(x3))) :|: z = 1 + (1 + x3), x3 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, y, 1 + x) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(x1))) :|: x1 >= 0, z = 1, z' = 1 + x1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(x''))) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(x'), double(exp(x2))) :|: z' = 1 + x2, z'' = 1 + (1 + x'), z = 1, x' >= 0, x2 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(x'), 1 + 0) :|: z'' = 1 + (1 + x'), z = 1, x' >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) tower(z) -{ 1 }-> f(0, x, 1 + 0) :|: x >= 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: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { double } { half } { exp } { f } { tower } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {tower} ---------------------------------------- (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: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {tower} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {tower} Previous analysis results are: double: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {half}, {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (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: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {half}, {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {half}, {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] half: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0 f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] half: runtime: O(n^1) [4 + z], size: O(n^1) [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: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0 half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] half: runtime: O(n^1) [4 + z], size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: exp after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0 half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] half: runtime: O(n^1) [4 + z], size: O(n^1) [z] exp: runtime: ?, size: EXP ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: exp after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0 exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0 exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0 f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0 f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0 f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0 f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0 f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0 half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0 half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0 half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0 tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0 Function symbols to be analyzed: {exp}, {f}, {tower} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] half: runtime: O(n^1) [4 + z], size: O(n^1) [z] exp: runtime: INF, size: EXP ---------------------------------------- (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: none And the following fresh constants: none ---------------------------------------- (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: tower(x) -> f(a, x, s(0)) [1] f(a, 0, y) -> y [1] f(a, s(x), y) -> f(b, y, s(x)) [1] f(b, y, x) -> f(a, half(x), exp(y)) [1] exp(0) -> s(0) [1] exp(s(x)) -> double(exp(x)) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] half(0) -> double(0) [1] half(s(0)) -> half(0) [1] half(s(s(x))) -> s(half(x)) [1] The TRS has the following type information: tower :: 0:s -> 0:s f :: a:b -> 0:s -> 0:s -> 0:s a :: a:b s :: 0:s -> 0:s 0 :: 0:s b :: a:b half :: 0:s -> 0:s exp :: 0:s -> 0:s double :: 0:s -> 0:s 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: a => 0 0 => 0 b => 1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x exp(z) -{ 1 }-> double(exp(x)) :|: x >= 0, z = 1 + x exp(z) -{ 1 }-> 1 + 0 :|: z = 0 f(z, z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z = 0, z' = 0 f(z, z', z'') -{ 1 }-> f(1, y, 1 + x) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0 f(z, z', z'') -{ 1 }-> f(0, half(x), exp(y)) :|: z = 1, y >= 0, x >= 0, z'' = x, z' = y half(z) -{ 1 }-> half(0) :|: z = 1 + 0 half(z) -{ 1 }-> double(0) :|: z = 0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) tower(z) -{ 1 }-> f(0, x, 1 + 0) :|: x >= 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: tower(z0) -> f(a, z0, s(0)) f(a, 0, z0) -> z0 f(a, s(z0), z1) -> f(b, z1, s(z0)) f(b, z0, z1) -> f(a, half(z1), exp(z0)) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) Tuples: TOWER(z0) -> c(F(a, z0, s(0))) F(a, 0, z0) -> c1 F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(0) -> c5 EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(0) -> c7 DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(0) -> c9(DOUBLE(0)) HALF(s(0)) -> c10(HALF(0)) HALF(s(s(z0))) -> c11(HALF(z0)) S tuples: TOWER(z0) -> c(F(a, z0, s(0))) F(a, 0, z0) -> c1 F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(0) -> c5 EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(0) -> c7 DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(0) -> c9(DOUBLE(0)) HALF(s(0)) -> c10(HALF(0)) HALF(s(s(z0))) -> c11(HALF(z0)) K tuples:none Defined Rule Symbols: tower_1, f_3, exp_1, double_1, half_1 Defined Pair Symbols: TOWER_1, F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c_1, c1, c2_1, c3_2, c4_2, c5, c6_2, c7, c8_1, c9_1, c10_1, c11_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: TOWER(z0) -> c(F(a, z0, s(0))) Removed 5 trailing nodes: EXP(0) -> c5 DOUBLE(0) -> c7 HALF(0) -> c9(DOUBLE(0)) HALF(s(0)) -> c10(HALF(0)) F(a, 0, z0) -> c1 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: tower(z0) -> f(a, z0, s(0)) f(a, 0, z0) -> z0 f(a, s(z0), z1) -> f(b, z1, s(z0)) f(b, z0, z1) -> f(a, half(z1), exp(z0)) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) K tuples:none Defined Rule Symbols: tower_1, f_3, exp_1, double_1, half_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c3_2, c4_2, c6_2, c8_1, c11_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: tower(z0) -> f(a, z0, s(0)) f(a, 0, z0) -> z0 f(a, s(z0), z1) -> f(b, z1, s(z0)) f(b, z0, z1) -> f(a, half(z1), exp(z0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) K tuples:none Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c3_2, c4_2, c6_2, c8_1, c11_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) by F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0)) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0))) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0)) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0))) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0)) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0))) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) K tuples:none Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c4_2, c6_2, c8_1, c11_1, c3_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) K tuples:none Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1 ---------------------------------------- (51) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1 ---------------------------------------- (53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [2]x_3 + [2]x_2*x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [1] POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = x_1 POL(exp(x_1)) = 0 POL(half(x_1)) = [2] POL(s(x_1)) = 0 ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) by F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0)) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0)) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0)) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1 ---------------------------------------- (59) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, x0, 0) -> c(EXP(x0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, x0, 0) -> c(EXP(x0)) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1 ---------------------------------------- (61) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F(b, x0, 0) -> c(EXP(x0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1 ---------------------------------------- (63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, x1) -> c4(F(a, half(x1), s(0))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [1] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = x_1^2 POL(exp(x_1)) = 0 POL(half(x_1)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, EXP_1, DOUBLE_1, HALF_1 Compound Symbols: c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) by EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) by F(b, 0, 0) -> c3(F(a, double(0), s(0)), HALF(0)) F(b, 0, s(0)) -> c3(F(a, half(0), s(0)), HALF(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, 0) -> c3(F(a, double(0), s(0)), HALF(0)) F(b, 0, s(0)) -> c3(F(a, half(0), s(0)), HALF(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) by F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0)) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0)) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0)) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (79) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) by F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) F(b, x0, 0) -> c3(F(a, 0, exp(x0))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, 0) -> c3(F(a, 0, exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) K tuples: F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, x0, 0) -> c3(F(a, 0, exp(x0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) by F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [2]x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [2] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(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(c8(x_1)) = x_1 POL(double(x_1)) = x_1 POL(exp(x_1)) = 0 POL(half(x_1)) = [2] POL(s(x_1)) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) by F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1 ---------------------------------------- (91) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, s(x0), 0) -> c1(EXP(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, s(x0), 0) -> c1(EXP(s(x0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (93) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F(b, s(x0), 0) -> c1(EXP(s(x0))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (95) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) by F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0)) F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0)) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0)) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [2]x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(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(c8(x_1)) = x_1 POL(double(x_1)) = x_1 POL(exp(x_1)) = 0 POL(half(x_1)) = [2] POL(s(x_1)) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, x1) -> c4(F(a, half(x1), s(0))) by F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) F(b, x0, 0) -> c(F(a, double(0), exp(x0))) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c_1, c6_2, c6_1, c3_1, c1_1, c4_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, 0) -> c(F(a, double(0), exp(x0))) by F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) F(b, x0, 0) -> c(F(a, 0, exp(x0))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) F(b, x0, 0) -> c(F(a, 0, exp(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) K tuples: F(b, x0, 0) -> c(F(a, double(0), exp(x0))) F(b, 0, x1) -> c4(F(a, half(x1), s(0))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_2, c6_1, c3_1, c1_1, c4_1, c_1 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, x0, 0) -> c(F(a, 0, exp(x0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_2, c6_1, c3_1, c1_1, c4_1, c_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) by EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2 ---------------------------------------- (111) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, 0) -> c3(F(a, double(0), s(0))) by F(b, 0, 0) -> c3(F(a, 0, s(0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, 0, 0) -> c3(F(a, 0, s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, 0) -> c3(F(a, 0, s(0))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, 0, s(0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, 0, s(0)) -> c3(F(a, 0, s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, s(0)) -> c3(F(a, 0, s(0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) by F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) by F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) by F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, 0, 0) -> c3(F(a, double(0), s(0))) F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, 0) -> c3(F(a, double(0), s(0))) by F(b, 0, 0) -> c3(F(a, 0, s(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, 0, 0) -> c3(F(a, 0, s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, 0) -> c3(F(a, 0, s(0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) K tuples: F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) K tuples: F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) by F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, 0, s(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, 0, s(0)) -> c3(F(a, 0, s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, s(0)) -> c3(F(a, 0, s(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) by F(b, z0, s(0)) -> c3(F(a, 0, exp(z0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, z0, s(0)) -> c3(F(a, 0, exp(z0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, z0, s(0)) -> c3(F(a, 0, exp(z0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, z0, s(0)) -> c3(F(a, 0, exp(z0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) by F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (155) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) by F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (157) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) by F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) by F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) by F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0)))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) K tuples: F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c1_1 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0))) by F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) S tuples: F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: F_3, DOUBLE_1, HALF_1, EXP_1 Compound Symbols: c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) by F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, F_3, EXP_1 Compound Symbols: c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (169) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(s(z0)) -> double(exp(z0)) exp(0) -> s(0) And the Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = x_1 + x_3 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = x_1 POL(exp(x_1)) = 0 POL(half(x_1)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, F_3, EXP_1 Compound Symbols: c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) by F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1)))) F(b, s(z0), s(s(x1))) -> c3(F(a, s(half(x1)), double(exp(z0))), HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, F_3, EXP_1 Compound Symbols: c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (173) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [1] + x_1 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = 0 POL(exp(x_1)) = 0 POL(half(x_1)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, F_3, EXP_1 Compound Symbols: c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (175) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [2]x_3 + [2]x_1*x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [1] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = x_1^2 POL(exp(x_1)) = 0 POL(half(x_1)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, F_3, EXP_1 Compound Symbols: c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) by F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0)) F(b, s(z0), s(s(x1))) -> c4(F(a, s(half(x1)), double(exp(z0))), EXP(s(z0))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0)) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0)) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0))) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) We considered the (Usable) Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) half(s(s(z0))) -> s(half(z0)) exp(s(z0)) -> double(exp(z0)) half(s(0)) -> half(0) exp(0) -> s(0) half(0) -> double(0) And the Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [2] + [2]x_3 + x_1*x_2 POL(HALF(x_1)) = 0 POL(a) = 0 POL(b) = [1] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = [2]x_1 POL(exp(x_1)) = 0 POL(half(x_1)) = [2] POL(s(x_1)) = 0 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) by F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (187) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(b, z0, s(0)) -> c4(EXP(z0)) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(DOUBLE(x_1)) = 0 POL(EXP(x_1)) = 0 POL(F(x_1, x_2, x_3)) = [1] POL(HALF(x_1)) = 0 POL(a) = [3] POL(b) = [3] POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(double(x_1)) = [2] POL(exp(x_1)) = 0 POL(half(x_1)) = [2]x_1 POL(s(x_1)) = 0 ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, 0, s(0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, s(0)) -> c4(F(a, 0, s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, s(0)) -> c4(F(a, 0, s(0))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, 0) -> c4(F(a, double(0), s(0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, 0) -> c4(F(a, double(0), s(0))) by F(b, 0, 0) -> c4(F(a, 0, s(0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, 0) -> c4(F(a, 0, s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, 0) -> c4(F(a, 0, s(0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (199) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, 0, s(0))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, s(0)) -> c4(F(a, 0, s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, s(0)) -> c4(F(a, 0, s(0))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, 0, 0) -> c(F(a, double(0), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, 0, 0) -> c(F(a, double(0), s(0))) by F(b, 0, 0) -> c(F(a, 0, s(0))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) F(b, 0, 0) -> c(F(a, 0, s(0))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, 0, 0) -> c(F(a, 0, s(0))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: DOUBLE_1, HALF_1, EXP_1, F_3 Compound Symbols: c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1 ---------------------------------------- (209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(z0)) -> c8(DOUBLE(z0)) by DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) EXP(s(0)) -> c6(DOUBLE(s(0))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) EXP(s(s(0))) -> c5(EXP(s(0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, EXP_1, F_3, DOUBLE_1 Compound Symbols: c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1 ---------------------------------------- (211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: EXP(s(s(0))) -> c5(EXP(s(0))) EXP(s(0)) -> c6(DOUBLE(s(0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1 ---------------------------------------- (213) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) by F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0)))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0)))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c5_1, c3_1, c2_1, c8_1, c_1 ---------------------------------------- (217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0)))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c5_1, c3_1, c2_1, c8_1 ---------------------------------------- (219) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) by EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1 ---------------------------------------- (221) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1 ---------------------------------------- (223) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1 ---------------------------------------- (225) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1 ---------------------------------------- (227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0)))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1 ---------------------------------------- (229) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) by F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: half(0) -> double(0) half(s(0)) -> half(0) half(s(s(z0))) -> s(half(z0)) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) exp(0) -> s(0) exp(s(z0)) -> double(exp(z0)) Tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) F(b, z0, s(0)) -> c4(EXP(z0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0))) S tuples: HALF(s(s(z0))) -> c11(HALF(z0)) F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0)))) F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0))) EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0))) F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0)))) F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0))) F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1)))) F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1)))) F(a, s(0), z1) -> c2(F(b, z1, s(0))) F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0))) F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0))) F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0))) F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0)))) F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0))))) F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0)))))) F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0)) F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0)) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0)))) EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0))))) F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0))) K tuples: F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0)))) F(a, s(0), 0) -> c2(F(b, 0, s(0))) F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1)))) F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0)))) F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0))) F(b, z0, s(0)) -> c4(EXP(z0)) Defined Rule Symbols: half_1, double_1, exp_1 Defined Pair Symbols: HALF_1, F_3, EXP_1, DOUBLE_1 Compound Symbols: c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1