KILLED proof of input_vw0monivZx.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 1 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 81 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 450 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 323 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 283 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 162 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 974 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 686 ms] (58) CpxRNTS (59) CompletionProof [UPPER BOUND(ID), 0 ms] (60) CpxTypedWeightedCompleteTrs (61) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 5 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 147 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 136 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 206 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 187 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 22 ms] (124) CdtProblem (125) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 201 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 313 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 235 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 6 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 287 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 5 ms] (166) CdtProblem (167) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 277 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 5 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 264 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 276 ms] (190) CdtProblem (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 9 ms] (192) CdtProblem (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (208) CdtProblem (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 7 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 14 ms] (216) CdtProblem (217) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 224 ms] (220) CdtProblem (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 15 ms] (228) CdtProblem (229) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (234) CdtProblem (235) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 26 ms] (236) CdtProblem (237) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtRewritingProof [BOTH BOUNDS(ID, ID), 40 ms] (240) CdtProblem (241) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CdtProblem (243) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 215 ms] (244) CdtProblem (245) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CdtProblem (247) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (248) 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: min(0, y) -> 0 min(x, 0) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(0, y) -> y max(x, 0) -> x max(s(x), s(y)) -> s(max(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) p(s(x)) -> x f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(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: min(0', y) -> 0' min(x, 0') -> 0' min(s(x), s(y)) -> s(min(x, y)) max(0', y) -> y max(x, 0') -> x max(s(x), s(y)) -> s(max(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) p(s(x)) -> x f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(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: min(0, y) -> 0 min(x, 0) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(0, y) -> y max(x, 0) -> x max(s(x), s(y)) -> s(max(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) p(s(x)) -> x f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] p(s(x)) -> x [1] f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s p :: 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 max_2 min_2 p_1 twice_1 Due to the following rules being added: minus(v0, v1) -> 0 [0] p(v0) -> 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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1] minus(v0, v1) -> 0 [0] p(v0) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s p :: 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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] p(s(x)) -> x [1] f(s(0), s(y)) -> f(minus(s(max(0, y)), s(min(0, y))), p(twice(0))) [4] f(s(x), s(0)) -> f(minus(s(max(x, 0)), s(min(x, 0))), p(twice(0))) [4] f(s(s(x')), s(s(y'))) -> f(minus(s(max(s(x'), s(y'))), s(min(s(x'), s(y')))), p(twice(s(min(x', y'))))) [4] minus(v0, v1) -> 0 [0] p(v0) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s twice :: 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s p :: 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') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { twice } { max } { minus } { min } { p } { f } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f} ---------------------------------------- (25) 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 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f} Previous analysis results are: twice: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 twice(z) -{ 1 }-> 0 :|: z = 0 twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {max}, {minus}, {min}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {max}, {minus}, {min}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (37) 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 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (39) 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' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (43) 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 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: ?, size: O(n^1) [z] ---------------------------------------- (45) 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' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {f} Previous analysis results are: twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (55) 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 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] p: runtime: O(1) [1], size: O(n^1) [z] f: runtime: ?, size: O(1) [0] ---------------------------------------- (57) 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: ? ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0 f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0 f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), 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 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= 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 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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] max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + 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] p: runtime: O(1) [1], size: O(n^1) [z] f: runtime: INF, size: O(1) [0] ---------------------------------------- (59) 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] p(v0) -> null_p [0] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] twice(v0) -> null_twice [0] And the following fresh constants: null_minus, null_p, null_f, null_min, null_max, null_twice ---------------------------------------- (60) 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: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] twice(0) -> 0 [1] twice(s(x)) -> s(s(twice(x))) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1] minus(v0, v1) -> null_minus [0] p(v0) -> null_p [0] f(v0, v1) -> null_f [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] twice(v0) -> null_twice [0] The TRS has the following type information: min :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice 0 :: 0:s:null_minus:null_p:null_min:null_max:null_twice s :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice max :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice p :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice f :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> null_f null_minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice null_p :: 0:s:null_minus:null_p:null_min:null_max:null_twice null_f :: null_f null_min :: 0:s:null_minus:null_p:null_min:null_max:null_twice null_max :: 0:s:null_minus:null_p:null_min:null_max:null_twice null_twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice Rewrite Strategy: INNERMOST ---------------------------------------- (61) 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_p => 0 null_f => 0 null_min => 0 null_max => 0 null_twice => 0 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(minus(max(1 + x, 1 + y), min(1 + x, 1 + y)), p(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 max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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. ---------------------------------------- (63) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))) Tuples: MIN(0, z0) -> c MIN(z0, 0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(0, z0) -> c3 MAX(z0, 0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(0) -> c6 TWICE(s(z0)) -> c7(TWICE(z0)) -'(z0, 0) -> c8 -'(s(z0), s(z1)) -> c9(-'(z0, z1)) P(s(z0)) -> c10 F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(0, z0) -> c MIN(z0, 0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(0, z0) -> c3 MAX(z0, 0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(0) -> c6 TWICE(s(z0)) -> c7(TWICE(z0)) -'(z0, 0) -> c8 -'(s(z0), s(z1)) -> c9(-'(z0, z1)) P(s(z0)) -> c10 F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: min_2, max_2, twice_1, -_2, p_1, f_2 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, P_1, F_2 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_3, c12_3, c13_4 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: P(s(z0)) -> c10 MIN(z0, 0) -> c1 -'(z0, 0) -> c8 TWICE(0) -> c6 MAX(z0, 0) -> c4 MIN(0, z0) -> c MAX(0, z0) -> c3 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: min_2, max_2, twice_1, -_2, p_1, f_2 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_4 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: min_2, max_2, twice_1, -_2, p_1, f_2 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3 ---------------------------------------- (69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c11_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c11_3, c12_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) by F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) by F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c11_2 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) K tuples:none Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1 ---------------------------------------- (87) 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)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) by F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c11_2 ---------------------------------------- (93) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1 ---------------------------------------- (95) 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)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c11(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) by F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) by F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c12_2 ---------------------------------------- (107) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1 ---------------------------------------- (109) 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)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) by F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c12_2 ---------------------------------------- (115) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1 ---------------------------------------- (117) 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)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(z0)), s(s(z1))) -> c12(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1 ---------------------------------------- (125) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1 ---------------------------------------- (127) 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(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by F(s(s(z0)), s(s(z1))) -> c13(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0)) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0)) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0)) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0))) F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1))) F(s(0), s(z1)) -> c13(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), TWICE(min(0, z1))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1 ---------------------------------------- (141) 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)) -> c13(TWICE(min(0, x0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (147) 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)) -> c13(TWICE(min(x0, 0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c11_2 ---------------------------------------- (157) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1 ---------------------------------------- (159) 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(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_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) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c11_2 ---------------------------------------- (165) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1 ---------------------------------------- (167) 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(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MIN_2, MAX_2, TWICE_1, -'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) by MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(s(x0), s(0)) -> c4(MIN(s(x0), s(0))) F(s(0), s(x0)) -> c3(MIN(s(0), s(x0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) by F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2 ---------------------------------------- (177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(-'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(-'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c11_2 ---------------------------------------- (179) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1 ---------------------------------------- (181) 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(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) We considered the (Usable) Rules:none And the Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) by F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(-'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c11(-'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c11_2 ---------------------------------------- (187) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1 ---------------------------------------- (189) 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(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) We considered the (Usable) Rules:none And the Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) 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] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1 ---------------------------------------- (191) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) by F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1 ---------------------------------------- (193) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c11_2 ---------------------------------------- (195) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c16(MAX(s(0), s(z0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: MAX_2, TWICE_1, -'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) by MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c16(MAX(s(0), s(z0))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) S tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: TWICE_1, -'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1 ---------------------------------------- (199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: F(s(0), s(z0)) -> c16(MAX(s(0), s(z0))) F(s(0), s(z0)) -> c10(MAX(s(0), s(z0))) F(s(0), s(x0)) -> c(MAX(s(0), s(x0))) F(s(x0), s(0)) -> c1(MAX(s(x0), s(0))) F(s(0), s(z0)) -> c8(MAX(s(0), s(z0))) F(s(z0), s(0)) -> c15(MAX(s(z0), s(0))) F(s(z0), s(0)) -> c14(MAX(s(z0), s(0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) S tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: TWICE_1, -'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) S tuples: TWICE(s(z0)) -> c7(TWICE(z0)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: TWICE_1, -'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2 ---------------------------------------- (203) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TWICE(s(z0)) -> c7(TWICE(z0)) by TWICE(s(s(y0))) -> c7(TWICE(s(y0))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) S tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: -'_2, F_2, MIN_2, MAX_2, TWICE_1 Compound Symbols: c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1 ---------------------------------------- (205) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c9(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (207) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (209) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) by F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) by none ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (217) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (219) 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(z0)) -> c13(TWICE(min(0, z0))) We considered the (Usable) Rules:none And the Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) 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] POL(MAX(x_1, x_2)) = x_1 + x_2 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (221) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (223) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(z0), s(0)) -> c13(TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (225) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(z0), s(0)) -> c13(TWICE(min(z0, 0))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (227) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (229) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (231) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (233) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) by none ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MIN_2, MAX_2, TWICE_1, -'_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c2_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1 ---------------------------------------- (235) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) by MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (237) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (239) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (241) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) F(s(z0), s(0)) -> c13(TWICE(min(z0, 0))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (243) 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)) -> c13(TWICE(min(x0, 0))) We considered the (Usable) Rules:none And the Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) 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] POL(MAX(x_1, x_2)) = x_1 + x_2 POL(MIN(x_1, x_2)) = 0 POL(TWICE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = 0 POL(twice(x_1)) = [1] + x_1 ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (245) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0))) ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1 ---------------------------------------- (247) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 p(s(z0)) -> z0 twice(0) -> 0 twice(s(z0)) -> s(s(twice(z0))) Tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c16(-'(max(s(0), s(z0)), min(s(0), s(z0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) S tuples: F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1))))) F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1)))) F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(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)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0))) F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0))) F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0)))) F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0)))) TWICE(s(s(y0))) -> c7(TWICE(s(y0))) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1)))) F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0))))) F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0))))) K tuples: F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0)))) F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0)))) F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1)))) F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0)))) F(s(z0), s(0)) -> c14(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(z0), s(0)) -> c15(-'(max(s(z0), s(0)), min(s(z0), s(0)))) F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1)))) F(s(0), s(z0)) -> c13(TWICE(min(0, z0))) F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) Defined Rule Symbols: -_2, max_2, min_2, p_1, twice_1 Defined Pair Symbols: F_2, MAX_2, TWICE_1, -'_2, MIN_2 Compound Symbols: c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c8_1, c10_1, c12_2, c14_1, c15_1, c16_1, c5_1, c11_2, c7_1, c9_1, c2_1