KILLED proof of input_Bs0N1uAiNV.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 8 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 318 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 235 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 1752 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 837 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [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^1)), 116 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 159 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1209 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1194 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 142 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1134 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1123 ms] (180) CdtProblem (181) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1104 ms] (186) CdtProblem (187) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: f_2 (c) The following functions are completely defined: minus_2 min_2 twice_1 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(y)) -> f(minus(y, 0), s(twice(0))) [3] f(s(s(x')), s(s(y'))) -> f(minus(s(y'), s(min(x', y'))), s(twice(s(min(x', y'))))) [3] f(s(x), s(0)) -> f(minus(x, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3] f(s(0), s(y)) -> f(minus(0, 0), s(twice(0))) [3] f(s(s(x'')), s(s(y''))) -> f(minus(s(x''), s(min(x'', y''))), s(twice(s(min(x'', y''))))) [3] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s min :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(x, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 3 }-> f(minus(y, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 3 }-> f(minus(1 + x'', 1 + min(x'', y'')), 1 + twice(1 + min(x'', y''))) :|: z' = 1 + (1 + y''), y'' >= 0, x'' >= 0, z = 1 + (1 + x'') f(z, z') -{ 3 }-> f(minus(1 + y', 1 + min(x', y')), 1 + twice(1 + min(x', y'))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { twice } { minus } { min } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {minus}, {min}, {f} Previous analysis results are: twice: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: twice after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {minus}, {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {min}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] f: runtime: ?, size: O(1) [0] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0 f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 2 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] f: runtime: INF, size: O(1) [0] ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] And the following fresh constants: null_minus, null_f, null_min, null_twice ---------------------------------------- (46) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] min(x, 0) -> 0 [1] min(0, y) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] f(s(x), s(y)) -> f(minus(y, min(x, y)), s(twice(min(x, y)))) [1] f(s(x), s(y)) -> f(minus(x, min(x, y)), s(twice(min(x, y)))) [1] minus(v0, v1) -> null_minus [0] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] twice(v0) -> null_twice [0] The TRS has the following type information: minus :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice 0 :: 0:s:null_minus:null_min:null_twice s :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice min :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice twice :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice f :: 0:s:null_minus:null_min:null_twice -> 0:s:null_minus:null_min:null_twice -> null_f null_minus :: 0:s:null_minus:null_min:null_twice null_f :: null_f null_min :: 0:s:null_minus:null_min:null_twice null_twice :: 0:s:null_minus:null_min:null_twice Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_minus => 0 null_f => 0 null_min => 0 null_twice => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(minus(x, min(x, y)), 1 + twice(min(x, y))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x f(z, z') -{ 1 }-> f(minus(y, min(x, y)), 1 + twice(min(x, y))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(z0, 0) -> c2 MIN(0, z0) -> c3 MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(0) -> c5 TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1, f_2 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c, c1_1, c2, c3, c4_1, c5, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: MIN(z0, 0) -> c2 TWICE(0) -> c5 -'(z0, 0) -> c MIN(0, z0) -> c3 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1, f_2 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1)))) f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3, c7_1, c7_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) by F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) by F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) K tuples:none Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) by F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) by F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) by F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) by F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1 ---------------------------------------- (137) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = 0 ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: -'_2, MIN_2, TWICE_1, F_2 Compound Symbols: c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) S tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: MIN_2, TWICE_1, F_2, -'_2 Compound Symbols: c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: MIN_2, TWICE_1, F_2, -'_2 Compound Symbols: c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) by MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7(F(0, s(0))) S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7(F(0, s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7 S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7 K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7 ---------------------------------------- (153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0), s(0)) -> c7 We considered the (Usable) Rules:none And the Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7 The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] + x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_2 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7) = 0 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7 S tuples: TWICE(s(z0)) -> c6(TWICE(z0)) F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(x0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: TWICE_1, F_2, -'_2, MIN_2 Compound Symbols: c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TWICE(s(z0)) -> c6(TWICE(z0)) by TWICE(s(s(y0))) -> c6(TWICE(s(y0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) F(s(x0), s(0)) -> c7 TWICE(s(s(y0))) -> c6(TWICE(s(y0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(x0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7, c6_1 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7 ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7(F(0, s(0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(0), s(0)) -> c7 We considered the (Usable) Rules:none And the Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 The order we found is given by the following interpretation: Polynomial interpretation : POL(-(x_1, x_2)) = [1] + x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1, x_2)) = [1] + x_2 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(TWICE(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7) = 0 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(min(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) by F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(0)) -> c7 F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) F(s(0), s(0)) -> c7 Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7 ---------------------------------------- (169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7 ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(x0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(x0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(0, s(0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1 ---------------------------------------- (187) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7(F(0, s(0))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) min(z0, 0) -> 0 min(0, z0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0))) F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0))) F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0))) F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0))) F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0))) F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0))) F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0))) F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0)))) F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0))) F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0))) F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0))) F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0)))) F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0))))) F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0)))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0))))) F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0)))))) TWICE(s(s(y0))) -> c6(TWICE(s(y0))) F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0))) K tuples: F(s(x0), s(0)) -> c8(TWICE(min(x0, 0))) F(s(0), s(x0)) -> c10(TWICE(min(0, x0))) Defined Rule Symbols: -_2, min_2, twice_1 Defined Pair Symbols: F_2, -'_2, MIN_2, TWICE_1 Compound Symbols: c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1