KILLED proof of input_ugqEuNiWCb.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 30 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 231 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 317 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 188 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 120 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 99 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 343 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 556 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 339 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 1713 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 610 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 18.9 s] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 6575 ms] (72) CpxRNTS (73) CompletionProof [UPPER BOUND(ID), 0 ms] (74) CpxTypedWeightedCompleteTrs (75) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms] (76) CpxRNTS (77) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 7 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 3 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 164 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 185 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 181 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 200 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 185 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 236 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (162) CdtProblem (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 240 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (166) CdtProblem (167) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 264 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 233 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 260 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 365 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 351 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 445 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 359 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 401 ms] (228) CdtProblem (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 409 ms] (240) CdtProblem (241) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (242) CdtProblem (243) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (244) CdtProblem (245) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 487 ms] (246) CdtProblem (247) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (248) CdtProblem (249) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CdtProblem (251) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 373 ms] (252) CdtProblem (253) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (254) CdtProblem (255) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (256) CdtProblem (257) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 427 ms] (258) CdtProblem (259) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (260) CdtProblem (261) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (262) CdtProblem (263) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (264) CdtProblem (265) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (266) CdtProblem (267) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (268) CdtProblem (269) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 453 ms] (270) CdtProblem (271) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (272) 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: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) min(add(n, nil)) -> n min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) if_min(true, add(n, add(m, x))) -> min(add(n, x)) if_min(false, add(n, add(m, x))) -> min(add(m, x)) head(add(n, x)) -> n tail(add(n, x)) -> x tail(nil) -> nil null(nil) -> true null(add(n, x)) -> false rm(n, nil) -> nil rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) if_rm(true, n, add(m, x)) -> rm(n, x) if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) minsort(x) -> mins(x, nil, nil) mins(x, y, z) -> if(null(x), x, y, z) if(true, x, y, z) -> z if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 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: eq(0', 0') -> true eq(0', s(x)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) min(add(n, nil)) -> n min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) if_min(true, add(n, add(m, x))) -> min(add(n, x)) if_min(false, add(n, add(m, x))) -> min(add(m, x)) head(add(n, x)) -> n tail(add(n, x)) -> x tail(nil) -> nil null(nil) -> true null(add(n, x)) -> false rm(n, nil) -> nil rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) if_rm(true, n, add(m, x)) -> rm(n, x) if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) minsort(x) -> mins(x, nil, nil) mins(x, y, z) -> if(null(x), x, y, z) if(true, x, y, z) -> z if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 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: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) min(add(n, nil)) -> n min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) if_min(true, add(n, add(m, x))) -> min(add(n, x)) if_min(false, add(n, add(m, x))) -> min(add(m, x)) head(add(n, x)) -> n tail(add(n, x)) -> x tail(nil) -> nil null(nil) -> true null(add(n, x)) -> false rm(n, nil) -> nil rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) if_rm(true, n, add(m, x)) -> rm(n, x) if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) minsort(x) -> mins(x, nil, nil) mins(x, y, z) -> if(null(x), x, y, z) if(true, x, y, z) -> z if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] min(add(n, nil)) -> n [1] min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) [1] if_min(true, add(n, add(m, x))) -> min(add(n, x)) [1] if_min(false, add(n, add(m, x))) -> min(add(m, x)) [1] head(add(n, x)) -> n [1] tail(add(n, x)) -> x [1] tail(nil) -> nil [1] null(nil) -> true [1] null(add(n, x)) -> false [1] rm(n, nil) -> nil [1] rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) [1] if_rm(true, n, add(m, x)) -> rm(n, x) [1] if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) [1] minsort(x) -> mins(x, nil, nil) [1] mins(x, y, z) -> if(null(x), x, y, z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) [1] if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) [1] if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] min(add(n, nil)) -> n [1] min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) [1] if_min(true, add(n, add(m, x))) -> min(add(n, x)) [1] if_min(false, add(n, add(m, x))) -> min(add(m, x)) [1] head(add(n, x)) -> n [1] tail(add(n, x)) -> x [1] tail(nil) -> nil [1] null(nil) -> true [1] null(add(n, x)) -> false [1] rm(n, nil) -> nil [1] rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) [1] if_rm(true, n, add(m, x)) -> rm(n, x) [1] if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) [1] minsort(x) -> mins(x, nil, nil) [1] mins(x, y, z) -> if(null(x), x, y, z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) [1] if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) [1] if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false le :: 0:s -> 0:s -> true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add min :: nil:add -> 0:s if_min :: true:false -> nil:add -> 0:s head :: nil:add -> 0:s tail :: nil:add -> nil:add null :: nil:add -> true:false rm :: 0:s -> nil:add -> nil:add if_rm :: true:false -> 0:s -> nil:add -> nil:add minsort :: nil:add -> nil:add mins :: nil:add -> nil:add -> nil:add -> nil:add if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add if2 :: true:false -> nil:add -> nil:add -> nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: minsort_1 mins_3 if_4 if2_4 (c) The following functions are completely defined: eq_2 head_1 min_1 le_2 app_2 rm_2 tail_1 null_1 if_min_2 if_rm_3 Due to the following rules being added: head(v0) -> 0 [0] min(v0) -> 0 [0] if_min(v0, v1) -> 0 [0] if_rm(v0, v1, v2) -> nil [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] min(add(n, nil)) -> n [1] min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) [1] if_min(true, add(n, add(m, x))) -> min(add(n, x)) [1] if_min(false, add(n, add(m, x))) -> min(add(m, x)) [1] head(add(n, x)) -> n [1] tail(add(n, x)) -> x [1] tail(nil) -> nil [1] null(nil) -> true [1] null(add(n, x)) -> false [1] rm(n, nil) -> nil [1] rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) [1] if_rm(true, n, add(m, x)) -> rm(n, x) [1] if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) [1] minsort(x) -> mins(x, nil, nil) [1] mins(x, y, z) -> if(null(x), x, y, z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) [1] if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) [1] if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) [1] head(v0) -> 0 [0] min(v0) -> 0 [0] if_min(v0, v1) -> 0 [0] if_rm(v0, v1, v2) -> nil [0] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false le :: 0:s -> 0:s -> true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add min :: nil:add -> 0:s if_min :: true:false -> nil:add -> 0:s head :: nil:add -> 0:s tail :: nil:add -> nil:add null :: nil:add -> true:false rm :: 0:s -> nil:add -> nil:add if_rm :: true:false -> 0:s -> nil:add -> nil:add minsort :: nil:add -> nil:add mins :: nil:add -> nil:add -> nil:add -> nil:add if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add if2 :: true:false -> nil:add -> nil:add -> nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] min(add(n, nil)) -> n [1] min(add(0, add(m, x))) -> if_min(true, add(0, add(m, x))) [2] min(add(s(x'), add(0, x))) -> if_min(false, add(s(x'), add(0, x))) [2] min(add(s(x''), add(s(y'), x))) -> if_min(le(x'', y'), add(s(x''), add(s(y'), x))) [2] if_min(true, add(n, add(m, x))) -> min(add(n, x)) [1] if_min(false, add(n, add(m, x))) -> min(add(m, x)) [1] head(add(n, x)) -> n [1] tail(add(n, x)) -> x [1] tail(nil) -> nil [1] null(nil) -> true [1] null(add(n, x)) -> false [1] rm(n, nil) -> nil [1] rm(0, add(0, x)) -> if_rm(true, 0, add(0, x)) [2] rm(0, add(s(x1), x)) -> if_rm(false, 0, add(s(x1), x)) [2] rm(s(x2), add(0, x)) -> if_rm(false, s(x2), add(0, x)) [2] rm(s(x3), add(s(y''), x)) -> if_rm(eq(x3, y''), s(x3), add(s(y''), x)) [2] if_rm(true, n, add(m, x)) -> rm(n, x) [1] if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) [1] minsort(x) -> mins(x, nil, nil) [1] mins(nil, y, z) -> if(true, nil, y, z) [2] mins(add(n', x4), y, z) -> if(false, add(n', x4), y, z) [2] if(true, x, y, z) -> z [1] if(false, add(n'', nil), y, z) -> if2(eq(n'', n''), add(n'', nil), y, z) [3] if(false, add(n'', add(m', x6)), y, z) -> if2(eq(n'', if_min(le(n'', m'), add(n'', add(m', x6)))), add(n'', add(m', x6)), y, z) [3] if(false, add(n'', x5), y, z) -> if2(eq(n'', 0), add(n'', x5), y, z) [2] if(false, add(n1, nil), y, z) -> if2(eq(0, n1), add(n1, nil), y, z) [2] if(false, add(n2, add(m'', x7)), y, z) -> if2(eq(0, if_min(le(n2, m''), add(n2, add(m'', x7)))), add(n2, add(m'', x7)), y, z) [2] if(false, x, y, z) -> if2(eq(0, 0), x, y, z) [1] if2(true, add(n3, x8), y, z) -> mins(app(rm(n3, x8), y), nil, app(z, add(n3, nil))) [4] if2(true, add(n3, x8), y, z) -> mins(app(rm(n3, x8), y), nil, app(z, add(0, nil))) [3] if2(true, add(n4, x9), y, z) -> mins(app(rm(0, x9), y), nil, app(z, add(n4, nil))) [3] if2(true, add(n4, x9), y, z) -> mins(app(rm(0, x9), y), nil, app(z, add(0, nil))) [2] if2(true, nil, y, z) -> mins(app(rm(0, nil), y), nil, app(z, add(0, nil))) [2] if2(false, add(n5, x10), y, z) -> mins(x10, add(n5, y), z) [3] if2(false, add(n5, x10), y, z) -> mins(x10, add(0, y), z) [2] if2(false, nil, y, z) -> mins(nil, add(0, y), z) [2] head(v0) -> 0 [0] min(v0) -> 0 [0] if_min(v0, v1) -> 0 [0] if_rm(v0, v1, v2) -> nil [0] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false le :: 0:s -> 0:s -> true:false app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: 0:s -> nil:add -> nil:add min :: nil:add -> 0:s if_min :: true:false -> nil:add -> 0:s head :: nil:add -> 0:s tail :: nil:add -> nil:add null :: nil:add -> true:false rm :: 0:s -> nil:add -> nil:add if_rm :: true:false -> 0:s -> nil:add -> nil:add minsort :: nil:add -> nil:add mins :: nil:add -> nil:add -> nil:add -> nil:add if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add if2 :: true:false -> nil:add -> nil:add -> nil:add -> nil:add Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0 eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = 1 + x, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', n''), 1 + n'' + 0, y, z) :|: z1 = y, z >= 0, z2 = z, y >= 0, n'' >= 0, z'' = 1 + n'' + 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), y, z) :|: z1 = y, z >= 0, z2 = z, y >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, y, z) :|: z'' = 1 + n'' + x5, z1 = y, x5 >= 0, z >= 0, z2 = z, y >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, n1), 1 + n1 + 0, y, z) :|: z1 = y, z >= 0, z'' = 1 + n1 + 0, z2 = z, y >= 0, n1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), y, z) :|: z1 = y, z >= 0, m'' >= 0, x7 >= 0, z2 = z, y >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + y, z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + y, z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), y), 0, app(z, 1 + n3 + 0)) :|: z1 = y, z >= 0, x8 >= 0, z2 = z, y >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), y), 0, app(z, 1 + 0 + 0)) :|: z1 = y, z >= 0, x8 >= 0, z2 = z, y >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), y), 0, app(z, 1 + n4 + 0)) :|: z1 = y, n4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), y), 0, app(z, 1 + 0 + 0)) :|: z1 = y, n4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), y), 0, app(z, 1 + 0 + 0)) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + y, z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 if_rm(z', z'', z1) -{ 1 }-> rm(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 1 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0 min(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 mins(z', z'', z1) -{ 2 }-> if(1, 0, y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, y, z) :|: z1 = z, x4 >= 0, z >= 0, z' = 1 + n' + x4, z'' = y, y >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(x, 0, 0) :|: z' = x, x >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(x3, y''), 1 + x3, 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, z' = 1 + x3, x >= 0, y'' >= 0, x3 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + x) :|: x >= 0, z'' = 1 + 0 + x, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + x2, 1 + 0 + x) :|: z' = 1 + x2, x >= 0, z'' = 1 + 0 + x, x2 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, n >= 0, z' = n tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { eq } { tail } { head } { null } { app } { min, if_min } { if_rm, rm } { if2, mins, if } { minsort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: ?, size: O(1) [1] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z'' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> if2(eq(n'', if_min(le(n'', m'), 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, if_min(le(n2, m''), 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + x)) :|: x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(n'', 0), 1 + n'' + x5, z1, z2) :|: z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 1 }-> if2(eq(0, 0), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 2 }-> if2(eq(0, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 3 }-> if2(eq(z'' - 1, z'' - 1), 1 + (z'' - 1) + 0, z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 2 }-> if_rm(eq(z' - 1, y''), 1 + (z' - 1), 1 + (1 + y'') + x) :|: z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: null after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {null}, {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: ?, size: O(1) [1] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: null after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: app after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {app}, {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: app after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + n3 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(n3, x8), z1), 0, app(z2, 1 + 0 + 0)) :|: z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 3 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + n4 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, x9), z1), 0, app(z2, 1 + 0 + 0)) :|: n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 2 }-> mins(app(rm(0, 0), z1), 0, app(z2, 1 + 0 + 0)) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (57) 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' Computed SIZE bound using KoAT for: if_min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {min,if_min}, {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: ?, size: O(n^1) [z'] if_min: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 5 + 4*z' + z'^2 Computed RUNTIME bound using KoAT for: if_min after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 22 + 24*z'' + 8*z''^2 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + m' }-> if2(eq(n'', if_min(s'', 1 + n'' + (1 + m' + x6))), 1 + n'' + (1 + m' + x6), z1, z2) :|: s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 4 + m'' }-> if2(eq(0, if_min(s1, 1 + n2 + (1 + m'' + x7))), 1 + n2 + (1 + m'' + x7), z1, z2) :|: s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + y' }-> if_min(s', 1 + (1 + x'') + (1 + (1 + y') + x)) :|: s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 2 }-> if_min(1, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(0, 1 + (1 + x') + (1 + 0 + x)) :|: x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: if_rm after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z1 Computed SIZE bound using CoFloCo for: rm after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if_rm,rm}, {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] if_rm: runtime: ?, size: O(n^1) [z1] rm: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if_rm after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 12 + 2*z'' + z''*z1 + 5*z1 Computed RUNTIME bound using KoAT for: rm after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 79 + 7*z' + 3*z'*z'' + 30*z'' ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(n3, x8), z1), 0, s10) :|: s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 5 + z2 }-> mins(app(rm(n3, x8), z1), 0, s9) :|: s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 4 + z2 }-> mins(app(rm(0, x9), z1), 0, s11) :|: s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, x9), z1), 0, s12) :|: s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 3 + z2 }-> mins(app(rm(0, 0), z1), 0, s13) :|: s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 1 }-> rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 4 + z' }-> if_rm(s3, 1 + (z' - 1), 1 + (1 + y'') + x) :|: s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 2 }-> if_rm(1, 0, 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 2 }-> if_rm(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 2 }-> if_rm(0, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] if_rm: runtime: O(n^2) [12 + 2*z'' + z''*z1 + 5*z1], size: O(n^1) [z1] rm: runtime: O(n^2) [79 + 7*z' + 3*z'*z'' + 30*z''], size: O(n^1) [z''] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 85 + 7*n3 + 3*n3*x8 + s29 + 30*x8 + z2 }-> mins(s30, 0, s9) :|: s29 >= 0, s29 <= x8, s30 >= 0, s30 <= s29 + z1, s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + 7*n3 + 3*n3*x8 + s31 + 30*x8 + z2 }-> mins(s32, 0, s10) :|: s31 >= 0, s31 <= x8, s32 >= 0, s32 <= s31 + z1, s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + s33 + 30*x9 + z2 }-> mins(s34, 0, s11) :|: s33 >= 0, s33 <= x9, s34 >= 0, s34 <= s33 + z1, s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s35 + 30*x9 + z2 }-> mins(s36, 0, s12) :|: s35 >= 0, s35 <= x9, s36 >= 0, s36 <= s35 + z1, s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s37 + z2 }-> mins(s38, 0, s13) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= s37 + z1, s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> s27 :|: s27 >= 0, s27 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> 1 + m + s28 :|: s28 >= 0, s28 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 14 + 5*z'' }-> s23 :|: s23 >= 0, s23 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 24 + 5*x + 5*x1 }-> s24 :|: s24 >= 0, s24 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 14 + 2*z' + z'*z'' + 5*z'' }-> s25 :|: s25 >= 0, s25 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 26 + 5*x + x*z' + 5*y'' + y''*z' + 5*z' }-> s26 :|: s26 >= 0, s26 <= 1 + (1 + y'') + x, s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] if_rm: runtime: O(n^2) [12 + 2*z'' + z''*z1 + 5*z1], size: O(n^1) [z1] rm: runtime: O(n^2) [79 + 7*z' + 3*z'*z'' + 30*z''], size: O(n^1) [z''] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 + z2 Computed SIZE bound using CoFloCo for: mins after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1 Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 + z2 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 85 + 7*n3 + 3*n3*x8 + s29 + 30*x8 + z2 }-> mins(s30, 0, s9) :|: s29 >= 0, s29 <= x8, s30 >= 0, s30 <= s29 + z1, s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + 7*n3 + 3*n3*x8 + s31 + 30*x8 + z2 }-> mins(s32, 0, s10) :|: s31 >= 0, s31 <= x8, s32 >= 0, s32 <= s31 + z1, s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + s33 + 30*x9 + z2 }-> mins(s34, 0, s11) :|: s33 >= 0, s33 <= x9, s34 >= 0, s34 <= s33 + z1, s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s35 + 30*x9 + z2 }-> mins(s36, 0, s12) :|: s35 >= 0, s35 <= x9, s36 >= 0, s36 <= s35 + z1, s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s37 + z2 }-> mins(s38, 0, s13) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= s37 + z1, s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> s27 :|: s27 >= 0, s27 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> 1 + m + s28 :|: s28 >= 0, s28 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 14 + 5*z'' }-> s23 :|: s23 >= 0, s23 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 24 + 5*x + 5*x1 }-> s24 :|: s24 >= 0, s24 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 14 + 2*z' + z'*z'' + 5*z'' }-> s25 :|: s25 >= 0, s25 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 26 + 5*x + x*z' + 5*y'' + y''*z' + 5*z' }-> s26 :|: s26 >= 0, s26 <= 1 + (1 + y'') + x, s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] if_rm: runtime: O(n^2) [12 + 2*z'' + z''*z1 + 5*z1], size: O(n^1) [z1] rm: runtime: O(n^2) [79 + 7*z' + 3*z'*z'' + 30*z''], size: O(n^1) [z''] if2: runtime: ?, size: O(n^1) [1 + z'' + z1 + z2] mins: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] if: runtime: ?, size: O(n^1) [1 + z'' + z1 + z2] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 app(z', z'') -{ 2 + x }-> 1 + n + s8 :|: s8 >= 0, s8 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0 eq(z', z'') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 110 + 57*m' + 16*m'*n'' + 16*m'*x6 + 8*m'^2 + 57*n'' + 16*n''*x6 + 8*n''^2 + 56*x6 + 8*x6^2 }-> if2(s20, 1 + n'' + (1 + m' + x6), z1, z2) :|: s19 >= 0, s19 <= 1 + n'' + (1 + m' + x6), s20 >= 0, s20 <= 1, s'' >= 0, s'' <= 1, z2 >= 0, z1 >= 0, m' >= 0, x6 >= 0, n'' >= 0, z'' = 1 + n'' + (1 + m' + x6), z' = 0 if(z', z'', z1, z2) -{ 109 + 57*m'' + 16*m''*n2 + 16*m''*x7 + 8*m''^2 + 56*n2 + 16*n2*x7 + 8*n2^2 + 56*x7 + 8*x7^2 }-> if2(s22, 1 + n2 + (1 + m'' + x7), z1, z2) :|: s21 >= 0, s21 <= 1 + n2 + (1 + m'' + x7), s22 >= 0, s22 <= 1, s1 >= 0, s1 <= 1, z2 >= 0, m'' >= 0, x7 >= 0, z1 >= 0, z'' = 1 + n2 + (1 + m'' + x7), n2 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + z'' }-> if2(s4, 1 + (z'' - 1) + 0, z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 + n'' }-> if2(s5, 1 + n'' + x5, z1, z2) :|: s5 >= 0, s5 <= 1, z'' = 1 + n'' + x5, x5 >= 0, z2 >= 0, z1 >= 0, n'' >= 0, z' = 0 if(z', z'', z1, z2) -{ 5 }-> if2(s6, 1 + (z'' - 1) + 0, z1, z2) :|: s6 >= 0, s6 <= 1, z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0 if(z', z'', z1, z2) -{ 4 }-> if2(s7, z'', z1, z2) :|: s7 >= 0, s7 <= 1, z2 >= 0, z'' >= 0, z1 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 85 + 7*n3 + 3*n3*x8 + s29 + 30*x8 + z2 }-> mins(s30, 0, s9) :|: s29 >= 0, s29 <= x8, s30 >= 0, s30 <= s29 + z1, s9 >= 0, s9 <= z2 + (1 + n3 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + 7*n3 + 3*n3*x8 + s31 + 30*x8 + z2 }-> mins(s32, 0, s10) :|: s31 >= 0, s31 <= x8, s32 >= 0, s32 <= s31 + z1, s10 >= 0, s10 <= z2 + (1 + 0 + 0), z2 >= 0, x8 >= 0, z1 >= 0, z'' = 1 + n3 + x8, z' = 1, n3 >= 0 if2(z', z'', z1, z2) -{ 84 + s33 + 30*x9 + z2 }-> mins(s34, 0, s11) :|: s33 >= 0, s33 <= x9, s34 >= 0, s34 <= s33 + z1, s11 >= 0, s11 <= z2 + (1 + n4 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s35 + 30*x9 + z2 }-> mins(s36, 0, s12) :|: s35 >= 0, s35 <= x9, s36 >= 0, s36 <= s35 + z1, s12 >= 0, s12 <= z2 + (1 + 0 + 0), n4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n4 + x9, z' = 1, x9 >= 0 if2(z', z'', z1, z2) -{ 83 + s37 + z2 }-> mins(s38, 0, s13) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= s37 + z1, s13 >= 0, s13 <= z2 + (1 + 0 + 0), z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if2(z', z'', z1, z2) -{ 3 }-> mins(x10, 1 + n5 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(x10, 1 + 0 + z1, z2) :|: z2 >= 0, z1 >= 0, z'' = 1 + n5 + x10, n5 >= 0, x10 >= 0, z' = 0 if2(z', z'', z1, z2) -{ 2 }-> mins(0, 1 + 0 + z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s17 :|: s17 >= 0, s17 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s18 :|: s18 >= 0, s18 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> s27 :|: s27 >= 0, s27 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 if_rm(z', z'', z1) -{ 80 + 30*x + 3*x*z'' + 7*z'' }-> 1 + m + s28 :|: s28 >= 0, s28 <= x, z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 0, m >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s14 :|: s14 >= 0, s14 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*x + 16*x*x' + 8*x^2 + 72*x' + 8*x'^2 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + x') + (1 + 0 + x), x' >= 0, x >= 0, z' = 1 + (1 + x') + (1 + 0 + x) min(z') -{ 250 + 88*x + 16*x*x'' + 16*x*y' + 8*x^2 + 88*x'' + 16*x''*y' + 8*x''^2 + 89*y' + 8*y'^2 }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + (1 + (1 + y') + x), s' >= 0, s' <= 1, x >= 0, y' >= 0, z' = 1 + (1 + x'') + (1 + (1 + y') + x), x'' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 mins(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 mins(z', z'', z1) -{ 2 }-> if(0, 1 + n' + x4, z'', z1) :|: x4 >= 0, z1 >= 0, z' = 1 + n' + x4, z'' >= 0, n' >= 0 minsort(z') -{ 1 }-> mins(z', 0, 0) :|: z' >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0 rm(z', z'') -{ 14 + 5*z'' }-> s23 :|: s23 >= 0, s23 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' = 0 rm(z', z'') -{ 24 + 5*x + 5*x1 }-> s24 :|: s24 >= 0, s24 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 0, z'' = 1 + (1 + x1) + x rm(z', z'') -{ 14 + 2*z' + z'*z'' + 5*z'' }-> s25 :|: s25 >= 0, s25 <= 1 + 0 + (z'' - 1), z'' - 1 >= 0, z' - 1 >= 0 rm(z', z'') -{ 26 + 5*x + x*z' + 5*y'' + y''*z' + 5*z' }-> s26 :|: s26 >= 0, s26 <= 1 + (1 + y'') + x, s3 >= 0, s3 <= 1, z'' = 1 + (1 + y'') + x, x >= 0, y'' >= 0, z' - 1 >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {if2,mins,if}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] null: runtime: O(1) [1], size: O(1) [1] app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] if_rm: runtime: O(n^2) [12 + 2*z'' + z''*z1 + 5*z1], size: O(n^1) [z1] rm: runtime: O(n^2) [79 + 7*z' + 3*z'*z'' + 30*z''], size: O(n^1) [z''] if2: runtime: INF, size: O(n^1) [1 + z'' + z1 + z2] mins: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] if: runtime: ?, size: O(n^1) [1 + z'' + z1 + z2] ---------------------------------------- (73) 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: min(v0) -> null_min [0] if_min(v0, v1) -> null_if_min [0] head(v0) -> null_head [0] if_rm(v0, v1, v2) -> null_if_rm [0] eq(v0, v1) -> null_eq [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] tail(v0) -> null_tail [0] null(v0) -> null_null [0] rm(v0, v1) -> null_rm [0] if(v0, v1, v2, v3) -> null_if [0] if2(v0, v1, v2, v3) -> null_if2 [0] And the following fresh constants: null_min, null_if_min, null_head, null_if_rm, null_eq, null_le, null_app, null_tail, null_null, null_rm, null_if, null_if2 ---------------------------------------- (74) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] min(add(n, nil)) -> n [1] min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) [1] if_min(true, add(n, add(m, x))) -> min(add(n, x)) [1] if_min(false, add(n, add(m, x))) -> min(add(m, x)) [1] head(add(n, x)) -> n [1] tail(add(n, x)) -> x [1] tail(nil) -> nil [1] null(nil) -> true [1] null(add(n, x)) -> false [1] rm(n, nil) -> nil [1] rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) [1] if_rm(true, n, add(m, x)) -> rm(n, x) [1] if_rm(false, n, add(m, x)) -> add(m, rm(n, x)) [1] minsort(x) -> mins(x, nil, nil) [1] mins(x, y, z) -> if(null(x), x, y, z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z) [1] if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil))) [1] if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z) [1] min(v0) -> null_min [0] if_min(v0, v1) -> null_if_min [0] head(v0) -> null_head [0] if_rm(v0, v1, v2) -> null_if_rm [0] eq(v0, v1) -> null_eq [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] tail(v0) -> null_tail [0] null(v0) -> null_null [0] rm(v0, v1) -> null_rm [0] if(v0, v1, v2, v3) -> null_if [0] if2(v0, v1, v2, v3) -> null_if2 [0] The TRS has the following type information: eq :: 0:s:null_min:null_if_min:null_head -> 0:s:null_min:null_if_min:null_head -> true:false:null_eq:null_le:null_null 0 :: 0:s:null_min:null_if_min:null_head true :: true:false:null_eq:null_le:null_null s :: 0:s:null_min:null_if_min:null_head -> 0:s:null_min:null_if_min:null_head false :: true:false:null_eq:null_le:null_null le :: 0:s:null_min:null_if_min:null_head -> 0:s:null_min:null_if_min:null_head -> true:false:null_eq:null_le:null_null app :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 nil :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 add :: 0:s:null_min:null_if_min:null_head -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 min :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> 0:s:null_min:null_if_min:null_head if_min :: true:false:null_eq:null_le:null_null -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> 0:s:null_min:null_if_min:null_head head :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> 0:s:null_min:null_if_min:null_head tail :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> true:false:null_eq:null_le:null_null rm :: 0:s:null_min:null_if_min:null_head -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 if_rm :: true:false:null_eq:null_le:null_null -> 0:s:null_min:null_if_min:null_head -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 minsort :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 mins :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 if :: true:false:null_eq:null_le:null_null -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 if2 :: true:false:null_eq:null_le:null_null -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 -> nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_min :: 0:s:null_min:null_if_min:null_head null_if_min :: 0:s:null_min:null_if_min:null_head null_head :: 0:s:null_min:null_if_min:null_head null_if_rm :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_eq :: true:false:null_eq:null_le:null_null null_le :: true:false:null_eq:null_le:null_null null_app :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_tail :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_null :: true:false:null_eq:null_le:null_null null_rm :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_if :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 null_if2 :: nil:add:null_if_rm:null_app:null_tail:null_rm:null_if:null_if2 Rewrite Strategy: INNERMOST ---------------------------------------- (75) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 0 null_min => 0 null_if_min => 0 null_head => 0 null_if_rm => 0 null_eq => 0 null_le => 0 null_app => 0 null_tail => 0 null_null => 0 null_rm => 0 null_if => 0 null_if2 => 0 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0 eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: x >= 0, z'' = 1 + x, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 eq(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2) -{ 1 }-> if2(eq(head(x), min(x)), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 if2(z', z'', z1, z2) -{ 1 }-> mins(tail(x), 1 + head(x) + y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if2(z', z'', z1, z2) -{ 1 }-> mins(app(rm(head(x), tail(x)), y), 0, app(z, 1 + head(x) + 0)) :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x if2(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 if_rm(z', z'', z1) -{ 1 }-> rm(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0 if_rm(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 if_rm(z', z'', z1) -{ 1 }-> 1 + m + rm(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 min(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + 0 min(z') -{ 1 }-> if_min(le(n, m), 1 + n + (1 + m + x)) :|: n >= 0, x >= 0, z' = 1 + n + (1 + m + x), m >= 0 min(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 mins(z', z'', z1) -{ 1 }-> if(null(x), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0 minsort(z') -{ 1 }-> mins(x, 0, 0) :|: z' = x, x >= 0 null(z') -{ 1 }-> 2 :|: z' = 0 null(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 null(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 rm(z', z'') -{ 1 }-> if_rm(eq(n, m), n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 rm(z', z'') -{ 1 }-> 0 :|: z'' = 0, n >= 0, z' = n rm(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (77) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) head(add(z0, z1)) -> z0 tail(add(z0, z1)) -> z1 tail(nil) -> nil null(nil) -> true null(add(z0, z1)) -> false rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) minsort(z0) -> mins(z0, nil, nil) mins(z0, z1, z2) -> if(null(z0), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> if2(eq(head(z0), min(z0)), z0, z1, z2) if2(true, z0, z1, z2) -> mins(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))) if2(false, z0, z1, z2) -> mins(tail(z0), add(head(z0), z1), z2) Tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(nil, z0) -> c7 APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, nil)) -> c9 MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) HEAD(add(z0, z1)) -> c13 TAIL(add(z0, z1)) -> c14 TAIL(nil) -> c15 NULL(nil) -> c16 NULL(add(z0, z1)) -> c17 RM(z0, nil) -> c18 RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINSORT(z0) -> c22(MINS(z0, nil, nil)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2), NULL(z0)) IF(true, z0, z1, z2) -> c24 IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), HEAD(z0)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), TAIL(z0)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil)), HEAD(z0)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2), TAIL(z0)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2), HEAD(z0)) S tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(nil, z0) -> c7 APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, nil)) -> c9 MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) HEAD(add(z0, z1)) -> c13 TAIL(add(z0, z1)) -> c14 TAIL(nil) -> c15 NULL(nil) -> c16 NULL(add(z0, z1)) -> c17 RM(z0, nil) -> c18 RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINSORT(z0) -> c22(MINS(z0, nil, nil)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2), NULL(z0)) IF(true, z0, z1, z2) -> c24 IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), HEAD(z0)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), TAIL(z0)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil)), HEAD(z0)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2), TAIL(z0)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2), HEAD(z0)) K tuples:none Defined Rule Symbols: eq_2, le_2, app_2, min_1, if_min_2, head_1, tail_1, null_1, rm_2, if_rm_3, minsort_1, mins_3, if_4, if2_4 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, IF_MIN_2, HEAD_1, TAIL_1, NULL_1, RM_2, IF_RM_3, MINSORT_1, MINS_3, IF_4, IF2_4 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_1, c7, c8_1, c9, c10_2, c11_1, c12_1, c13, c14, c15, c16, c17, c18, c19_2, c20_1, c21_1, c22_1, c23_2, c24, c25_3, c26_3, c27_4, c28_4, c29_3, c30_2, c31_2 ---------------------------------------- (79) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: MINSORT(z0) -> c22(MINS(z0, nil, nil)) Removed 14 trailing nodes: TAIL(nil) -> c15 IF(true, z0, z1, z2) -> c24 NULL(nil) -> c16 EQ(0, 0) -> c LE(0, z0) -> c4 NULL(add(z0, z1)) -> c17 LE(s(z0), 0) -> c5 APP(nil, z0) -> c7 EQ(s(z0), 0) -> c2 EQ(0, s(z0)) -> c1 TAIL(add(z0, z1)) -> c14 RM(z0, nil) -> c18 HEAD(add(z0, z1)) -> c13 MIN(add(z0, nil)) -> c9 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) head(add(z0, z1)) -> z0 tail(add(z0, z1)) -> z1 tail(nil) -> nil null(nil) -> true null(add(z0, z1)) -> false rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) minsort(z0) -> mins(z0, nil, nil) mins(z0, z1, z2) -> if(null(z0), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> if2(eq(head(z0), min(z0)), z0, z1, z2) if2(true, z0, z1, z2) -> mins(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))) if2(false, z0, z1, z2) -> mins(tail(z0), add(head(z0), z1), z2) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2), NULL(z0)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), HEAD(z0)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), TAIL(z0)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil)), HEAD(z0)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2), TAIL(z0)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2), HEAD(z0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2), NULL(z0)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), HEAD(z0)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0)), TAIL(z0)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil)), HEAD(z0)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2), TAIL(z0)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2), HEAD(z0)) K tuples:none Defined Rule Symbols: eq_2, le_2, app_2, min_1, if_min_2, head_1, tail_1, null_1, rm_2, if_rm_3, minsort_1, mins_3, if_4, if2_4 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, IF_MIN_2, RM_2, IF_RM_3, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c10_2, c11_1, c12_1, c19_2, c20_1, c21_1, c23_2, c25_3, c26_3, c27_4, c28_4, c29_3, c30_2, c31_2 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) head(add(z0, z1)) -> z0 tail(add(z0, z1)) -> z1 tail(nil) -> nil null(nil) -> true null(add(z0, z1)) -> false rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) minsort(z0) -> mins(z0, nil, nil) mins(z0, z1, z2) -> if(null(z0), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> if2(eq(head(z0), min(z0)), z0, z1, z2) if2(true, z0, z1, z2) -> mins(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))) if2(false, z0, z1, z2) -> mins(tail(z0), add(head(z0), z1), z2) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) K tuples:none Defined Rule Symbols: eq_2, le_2, app_2, min_1, if_min_2, head_1, tail_1, null_1, rm_2, if_rm_3, minsort_1, mins_3, if_4, if2_4 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, IF_MIN_2, RM_2, IF_RM_3, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c10_2, c11_1, c12_1, c19_2, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1 ---------------------------------------- (83) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: minsort(z0) -> mins(z0, nil, nil) mins(z0, z1, z2) -> if(null(z0), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> if2(eq(head(z0), min(z0)), z0, z1, z2) if2(true, z0, z1, z2) -> mins(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))) if2(false, z0, z1, z2) -> mins(tail(z0), add(head(z0), z1), z2) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, IF_MIN_2, RM_2, IF_RM_3, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c10_2, c11_1, c12_1, c19_2, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(add(z0, add(z1, z2))) -> c10(IF_MIN(le(z0, z1), add(z0, add(z1, z2))), LE(z0, z1)) by MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2))), LE(0, z0)) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2))), LE(s(z0), 0)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2))), LE(0, z0)) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2))), LE(s(z0), 0)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2))), LE(0, z0)) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2))), LE(s(z0), 0)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, RM_2, IF_RM_3, IF_4, MINS_3, IF2_4, MIN_1 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c19_2, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, RM_2, IF_RM_3, IF_4, MINS_3, IF2_4, MIN_1 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c19_2, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(z0, add(z1, z2)) -> c19(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) by RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2)), EQ(0, 0)) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2)), EQ(0, s(z0))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2)), EQ(s(z0), 0)) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2)), EQ(0, 0)) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2)), EQ(0, s(z0))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2)), EQ(s(z0), 0)) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2)), EQ(0, 0)) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2)), EQ(0, s(z0))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2)), EQ(s(z0), 0)) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF_4, MINS_3, IF2_4, MIN_1, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF_4, MINS_3, IF2_4, MIN_1, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c26_3, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, z1, z2) -> c26(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0)), MIN(z0)) by IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MINS_3, IF_4, IF2_4, MIN_1, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MINS_3, IF_4, IF2_4, MIN_1, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c23_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINS(z0, z1, z2) -> c23(IF(null(z0), z0, z1, z2)) by MINS(nil, x1, x2) -> c23(IF(true, nil, x1, x2)) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(nil, x1, x2) -> c23(IF(true, nil, x1, x2)) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(nil, x1, x2) -> c23(IF(true, nil, x1, x2)) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF_4, IF2_4, MIN_1, RM_2, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MINS(nil, x1, x2) -> c23(IF(true, nil, x1, x2)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) null(nil) -> true null(add(z0, z1)) -> false app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) K tuples:none Defined Rule Symbols: le_2, eq_2, head_1, min_1, if_min_2, null_1, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF_4, IF2_4, MIN_1, RM_2, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1 ---------------------------------------- (101) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: null(nil) -> true null(add(z0, z1)) -> false ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) K tuples:none Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF_4, IF2_4, MIN_1, RM_2, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c25_2, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, z1, z2) -> c25(IF2(eq(head(z0), min(z0)), z0, z1, z2), EQ(head(z0), min(z0))) by IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) K tuples:none Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c27_3, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, z0, z1, z2) -> c27(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) by IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c27(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c27(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c27(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples:none Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c27(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c27(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1)) K tuples:none Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c27_2 ---------------------------------------- (109) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) K tuples:none Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1 ---------------------------------------- (111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1 ---------------------------------------- (113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c28_3, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, z0, z1, z2) -> c28(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(rm(head(z0), tail(z0)), z1), RM(head(z0), tail(z0))) by IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c28(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c28(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c28(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c28(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c28(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c28_2 ---------------------------------------- (119) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1 ---------------------------------------- (121) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c29_2, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, z0, z1, z2) -> c29(MINS(app(rm(head(z0), tail(z0)), z1), nil, app(z2, add(head(z0), nil))), APP(z2, add(head(z0), nil))) by IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(nil, add(head(x0), nil))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, x2) -> c29(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(x2, add(head(nil), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(nil, add(head(x0), nil))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, x2) -> c29(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(x2, add(head(nil), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(nil, add(head(x0), nil))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, x2) -> c29(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(x2, add(head(nil), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, x2) -> c29(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(x2, add(head(nil), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, x2) -> c29(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil))), APP(x2, add(head(nil), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1 ---------------------------------------- (129) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1 ---------------------------------------- (131) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1 ---------------------------------------- (133) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c30_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0, z1, z2) -> c30(MINS(tail(z0), add(head(z0), z1), z2)) by IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c30(MINS(nil, add(head(nil), x1), x2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c30(MINS(nil, add(head(nil), x1), x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c30(MINS(nil, add(head(nil), x1), x2)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF2(false, nil, x1, x2) -> c30(MINS(nil, add(head(nil), x1), x2)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, IF2_4, MIN_1, RM_2, IF_4, MINS_3 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c31_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0, z1, z2) -> c31(MINS(tail(z0), add(head(z0), z1), z2)) by IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c31(MINS(nil, add(head(nil), x1), x2)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c31(MINS(nil, add(head(nil), x1), x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, nil, x1, x2) -> c31(MINS(nil, add(head(nil), x1), x2)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF2(false, nil, x1, x2) -> c31(MINS(nil, add(head(nil), x1), x2)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_2, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(add(s(z0), add(s(z1), x2))) -> c10(IF_MIN(le(z0, z1), add(s(z0), add(s(z1), x2))), LE(s(z0), s(z1))) by MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_2, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(s(z0), add(s(z1), x2)) -> c19(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) by RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, add(z1, z2)), x1, x2) -> c26(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) by IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_1 ---------------------------------------- (149) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(IF2(x_1, x_2, x_3, x_4)) = [1] POL(IF_MIN(x_1, x_2)) = 0 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_3, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_1 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, z1), x1, x2) -> c26(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1))), MIN(add(z0, z1))) by IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(z0, add(z1, z2)), x2, x3) -> c26(IF2(eq(z0, if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x2, x3), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2)))), MIN(add(z0, add(z1, z2)))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil))), MIN(add(z0, nil))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1 ---------------------------------------- (155) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(IF2(x_1, x_2, x_3, x_4)) = x_1 POL(IF_MIN(x_1, x_2)) = 0 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = [1] ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, IF_4, MINS_3, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c26_2, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) by IF(false, add(z0, nil), x1, x2) -> c26(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2 ---------------------------------------- (159) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + [3]x_1 POL(IF2(x_1, x_2, x_3, x_4)) = [1] POL(IF_MIN(x_1, x_2)) = 0 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [3] + [3]x_2 POL(if_rm(x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = 0 POL(nil) = 0 POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(head(add(z0, nil)), z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) by IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_1 ---------------------------------------- (163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(IF2(x_1, x_2, x_3, x_4)) = [1] POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, add(z1, z2)), x1, x2) -> c25(IF2(eq(head(add(z0, add(z1, z2))), if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x1, x2), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) by IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_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. IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(IF2(x_1, x_2, x_3, x_4)) = x_1 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = [1] ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF_4, IF2_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c25_2, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_1 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, add(z0, z1), x1, x2) -> c25(IF2(eq(z0, min(add(z0, z1))), add(z0, z1), x1, x2), EQ(head(add(z0, z1)), min(add(z0, z1)))) by IF(false, add(z0, nil), x2, x3) -> c25(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, add(z1, z2)), x2, x3) -> c25(IF2(eq(z0, if_min(le(z0, z1), add(z0, add(z1, z2)))), add(z0, add(z1, z2)), x2, x3), EQ(head(add(z0, add(z1, z2))), min(add(z0, add(z1, z2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1 ---------------------------------------- (171) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(IF2(x_1, x_2, x_3, x_4)) = [1] POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) by IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1 ---------------------------------------- (175) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_2 ---------------------------------------- (177) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, add(z0, z1)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) by IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1 ---------------------------------------- (181) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c27(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c27_2 ---------------------------------------- (183) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_1 ---------------------------------------- (185) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c27(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(x0, nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_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. IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c27(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_1 ---------------------------------------- (193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c27_3, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c4_1, c5_1 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c27(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c27(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1 ---------------------------------------- (197) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) by IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) by IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) by IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil)), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_2 ---------------------------------------- (207) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, add(z0, z1)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) by IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1), RM(head(nil), tail(nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1 ---------------------------------------- (211) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c28(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c28_2 ---------------------------------------- (213) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c9_1 ---------------------------------------- (215) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c9_1 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c28(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(x0, nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, 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. IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c9_1 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c28(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c9_1 ---------------------------------------- (223) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c28_3, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c7_1, c9_1 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) by IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c28(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, x1), x2, nil) -> c28(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), RM(head(add(x0, x1)), tail(add(x0, x1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1 ---------------------------------------- (227) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1 ---------------------------------------- (229) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) by IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c1_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1 ---------------------------------------- (231) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) by IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1 ---------------------------------------- (233) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, add(z0, z1)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(z0, app(z1, add(head(x0), nil)))), APP(add(z0, z1), add(head(x0), nil))) by IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, nil, x1, add(x2, x3)) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, nil, x1, add(x2, x3)) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, nil, x1, add(x2, x3)) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil)))), APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1 ---------------------------------------- (235) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (237) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (239) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(IF2(x_1, x_2, x_3, x_4)) = x_1 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = [1] POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = [1] POL(false) = [1] POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = [1] ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (241) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, app(x2, add(z0, nil))), APP(x2, add(head(add(z0, z1)), nil))) by IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(x0, nil)))), APP(add(z0, z1), add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (243) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (245) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (247) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) by IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(add(z0, z1), add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (249) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (251) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_2, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c13_1 ---------------------------------------- (253) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, add(z0, z1), x1, x2) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, app(x2, add(head(add(z0, z1)), nil))), APP(x2, add(head(add(z0, z1)), nil))) by IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, add(z0, z1)) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(z0, app(z1, add(head(add(x0, x1)), nil)))), APP(add(z0, z1), add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) ---------------------------------------- (254) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil)), APP(nil, add(head(add(x0, x1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c13_1 ---------------------------------------- (255) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (256) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c13_1 ---------------------------------------- (257) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) app(nil, z0) -> z0 And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4)) = x_4 POL(IF_MIN(x_1, x_2)) = x_2 POL(IF_RM(x_1, x_2, x_3)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = x_1 POL(MINS(x_1, x_2, x_3)) = x_3 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = 0 POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = [1] POL(rm(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = 0 POL(true) = 0 ---------------------------------------- (258) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c29_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c13_1 ---------------------------------------- (259) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) by IF2(true, add(z0, z1), x1, nil) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil))) IF2(true, add(z0, z1), x1, nil) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, add(z0, z1), x1, nil) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil))) ---------------------------------------- (260) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1 ---------------------------------------- (261) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) by IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) ---------------------------------------- (262) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1 ---------------------------------------- (263) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, add(z0, z1), x1, x2) -> c30(MINS(tail(add(z0, z1)), add(z0, x1), x2)) by IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) ---------------------------------------- (264) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c30_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1 ---------------------------------------- (265) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, add(z0, z1), x1, x2) -> c30(MINS(z1, add(head(add(z0, z1)), x1), x2)) by IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) ---------------------------------------- (266) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1, c30_1 ---------------------------------------- (267) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, add(z0, z1), x1, x2) -> c31(MINS(tail(add(z0, z1)), add(z0, x1), x2)) by IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) ---------------------------------------- (268) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1, c30_1 ---------------------------------------- (269) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) We considered the (Usable) Rules: app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) app(nil, z0) -> z0 rm(z0, nil) -> nil tail(nil) -> nil tail(add(z0, z1)) -> z1 if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 POL(IF2(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(IF_MIN(x_1, x_2)) = 0 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINS(x_1, x_2, x_3)) = x_1 + x_2 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_2 POL(app(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c25(x_1, x_2)) = x_1 + x_2 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c28(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c29(x_1)) = x_1 POL(c29(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(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(eq(x_1, x_2)) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(if_min(x_1, x_2)) = [1] + x_2 POL(if_rm(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = [1] POL(nil) = 0 POL(rm(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(tail(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (270) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c31_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1, c30_1 ---------------------------------------- (271) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, add(z0, z1), x1, x2) -> c31(MINS(z1, add(head(add(z0, z1)), x1), x2)) by IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) ---------------------------------------- (272) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) head(add(z0, z1)) -> z0 min(add(z0, nil)) -> z0 min(add(z0, add(z1, z2))) -> if_min(le(z0, z1), add(z0, add(z1, z2))) if_min(true, add(z0, add(z1, z2))) -> min(add(z0, z2)) if_min(false, add(z0, add(z1, z2))) -> min(add(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) tail(add(z0, z1)) -> z1 tail(nil) -> nil if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c4(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c4(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c(APP(rm(head(nil), nil), x0)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, nil) -> c7(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x1, nil) -> c7(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x0, nil) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c1(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c1(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(true, nil, x0, x1) -> c1(APP(rm(head(nil), nil), x0)) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, nil, x1, nil) -> c29(MINS(app(rm(head(nil), nil), x1), nil, add(head(nil), nil))) IF2(true, nil, x0, nil) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(head(nil), nil))) IF2(true, nil, x0, add(z0, z1)) -> c2(MINS(app(rm(head(nil), nil), x0), nil, add(z0, app(z1, add(head(nil), nil))))) IF2(true, nil, x0, x1) -> c2(MINS(app(nil, x0), nil, app(x1, add(head(nil), nil)))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_MIN(true, add(z0, add(z1, z2))) -> c11(MIN(add(z0, z2))) IF_MIN(false, add(z0, add(z1, z2))) -> c12(MIN(add(z1, z2))) IF_RM(true, z0, add(z1, z2)) -> c20(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c21(RM(z0, z2)) MIN(add(0, add(z0, x2))) -> c10(IF_MIN(true, add(0, add(z0, x2)))) MIN(add(s(z0), add(0, x2))) -> c10(IF_MIN(false, add(s(z0), add(0, x2)))) RM(0, add(0, x2)) -> c19(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c19(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c19(IF_RM(false, s(z0), add(0, x2))) MINS(add(z0, z1), x1, x2) -> c23(IF(false, add(z0, z1), x1, x2)) MIN(add(s(0), add(s(z0), x2))) -> c10(IF_MIN(true, add(s(0), add(s(z0), x2))), LE(s(0), s(z0))) MIN(add(s(s(z0)), add(s(0), x2))) -> c10(IF_MIN(false, add(s(s(z0)), add(s(0), x2))), LE(s(s(z0)), s(0))) MIN(add(s(s(z0)), add(s(s(z1)), x2))) -> c10(IF_MIN(le(z0, z1), add(s(s(z0)), add(s(s(z1)), x2))), LE(s(s(z0)), s(s(z1)))) MIN(add(s(x0), add(s(x1), x2))) -> c10(LE(s(x0), s(x1))) RM(s(0), add(s(0), x2)) -> c19(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c19(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c19(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(s(z0)), add(s(s(z1)), x2)) -> c19(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), add(s(x1), x2)) -> c19(EQ(s(x0), s(x1))) IF(false, add(0, add(z0, x2)), x3, x4) -> c26(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2)))), MIN(add(0, add(z0, x2)))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2)))), MIN(add(s(z0), add(0, x2)))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c26(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2)))), MIN(add(s(z0), add(s(z1), x2)))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c26(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2)))), MIN(add(z0, add(x1, x2)))) IF(false, add(z0, nil), x2, x3) -> c26(IF2(eq(z0, z0), add(z0, nil), x2, x3), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(z0, nil), x1, x2) -> c25(IF2(eq(z0, z0), add(z0, nil), x1, x2), EQ(head(add(z0, nil)), min(add(z0, nil)))) IF(false, add(0, add(z0, x2)), x3, x4) -> c25(IF2(eq(head(add(0, add(z0, x2))), if_min(true, add(0, add(z0, x2)))), add(0, add(z0, x2)), x3, x4), EQ(head(add(0, add(z0, x2))), min(add(0, add(z0, x2))))) IF(false, add(s(z0), add(0, x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(0, x2))), if_min(false, add(s(z0), add(0, x2)))), add(s(z0), add(0, x2)), x3, x4), EQ(head(add(s(z0), add(0, x2))), min(add(s(z0), add(0, x2))))) IF(false, add(s(z0), add(s(z1), x2)), x3, x4) -> c25(IF2(eq(head(add(s(z0), add(s(z1), x2))), if_min(le(z0, z1), add(s(z0), add(s(z1), x2)))), add(s(z0), add(s(z1), x2)), x3, x4), EQ(head(add(s(z0), add(s(z1), x2))), min(add(s(z0), add(s(z1), x2))))) IF(false, add(z0, add(x1, x2)), x3, x4) -> c25(IF2(eq(z0, if_min(le(z0, x1), add(z0, add(x1, x2)))), add(z0, add(x1, x2)), x3, x4), EQ(head(add(z0, add(x1, x2))), min(add(z0, add(x1, x2))))) IF2(true, x0, x1, add(x2, nil)) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, x0, x1, add(x2, nil)) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(head(x0), nil))), APP(add(x2, nil), add(head(x0), nil))) IF2(true, x0, x1, add(x2, add(z0, z1))) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(x2, add(z0, app(z1, add(head(x0), nil))))), APP(add(x2, add(z0, z1)), add(head(x0), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(z0, nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, tail(add(z0, z1))), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(false, add(z0, z1), x2, x3) -> c30(MINS(z1, add(z0, x2), x3)) IF2(false, add(z0, z1), x2, x3) -> c31(MINS(z1, add(z0, x2), x3)) K tuples: IF2(true, nil, x1, x2) -> c(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c27(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c1(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c1(APP(rm(head(nil), tail(nil)), x1)) IF2(true, x0, x1, nil) -> c28(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil)), APP(rm(head(x0), tail(x0)), x1), RM(head(x0), tail(x0))) IF2(true, nil, x1, x2) -> c2(MINS(app(rm(head(nil), nil), x1), nil, app(x2, add(head(nil), nil)))) IF2(true, nil, x1, x2) -> c2(APP(x2, add(head(nil), nil))) IF2(true, x0, x1, nil) -> c29(MINS(app(rm(head(x0), tail(x0)), x1), nil, add(head(x0), nil))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c26(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c26(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF(false, add(x0, nil), x1, x2) -> c26(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, nil), x1, x2) -> c25(EQ(head(add(x0, nil)), min(add(x0, nil)))) IF(false, add(x0, add(x1, x2)), x3, x4) -> c25(EQ(head(add(x0, add(x1, x2))), min(add(x0, add(x1, x2))))) IF(false, add(x0, x1), x2, x3) -> c25(EQ(head(add(x0, x1)), min(add(x0, x1)))) IF2(true, nil, x1, add(x2, x3)) -> c5(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c5(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c27(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c9(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c9(APP(rm(head(nil), tail(nil)), x1)) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), nil, add(z0, nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, nil) -> c28(MINS(app(rm(z0, tail(add(z0, z1))), x1), nil, add(head(add(z0, z1)), nil)), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, nil, x1, add(x2, x3)) -> c13(MINS(app(rm(head(nil), nil), x1), nil, add(x2, app(x3, add(head(nil), nil))))) IF2(true, nil, x1, add(x2, x3)) -> c13(APP(add(x2, x3), add(head(nil), nil))) IF2(true, x0, x1, add(x2, x3)) -> c29(APP(add(x2, x3), add(head(x0), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), tail(add(x0, x1))), x2), nil, add(x0, nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(head(add(x0, x1)), x1), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(x0, x1), x2, nil) -> c29(MINS(app(rm(x0, tail(add(x0, x1))), x2), nil, add(head(add(x0, x1)), nil))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c27(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c27(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c27(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x1), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(x0, nil), x2, x3) -> c28(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(rm(head(add(x0, nil)), tail(add(x0, nil))), x2), RM(head(add(x0, nil)), tail(add(x0, nil)))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c28(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(rm(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), x2), RM(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))) IF2(true, add(z0, z1), x2, x3) -> c28(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(rm(head(add(z0, z1)), tail(add(z0, z1))), x2), RM(head(add(z0, z1)), tail(add(z0, z1)))) IF2(true, add(z0, z1), x1, add(x2, x3)) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x1), nil, add(x2, app(x3, add(head(add(z0, z1)), nil)))), APP(add(x2, x3), add(head(add(z0, z1)), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(head(add(z0, z1)), z1), x2), nil, app(x3, add(z0, nil))), APP(x3, add(head(add(z0, z1)), nil))) IF2(true, add(x0, nil), x2, x3) -> c29(MINS(app(nil, x2), nil, app(x3, add(head(add(x0, nil)), nil))), APP(x3, add(head(add(x0, nil)), nil))) IF2(true, add(x0, add(z1, z2)), x2, x3) -> c29(MINS(app(if_rm(eq(head(add(x0, add(z1, z2))), z1), head(add(x0, add(z1, z2))), add(z1, z2)), x2), nil, app(x3, add(head(add(x0, add(z1, z2))), nil))), APP(x3, add(head(add(x0, add(z1, z2))), nil))) IF2(true, add(z0, z1), x2, x3) -> c29(MINS(app(rm(z0, z1), x2), nil, app(x3, add(head(add(z0, z1)), nil))), APP(x3, add(head(add(z0, z1)), nil))) Defined Rule Symbols: eq_2, head_1, min_1, if_min_2, le_2, app_2, rm_2, tail_1, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, IF_MIN_2, IF_RM_3, MIN_1, RM_2, MINS_3, IF2_4, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_1, c12_1, c20_1, c21_1, c10_1, c19_1, c23_1, c2_1, c10_2, c19_2, c26_3, c26_1, c26_2, c25_2, c25_1, c27_3, c4_1, c5_1, c_1, c28_3, c7_1, c9_1, c1_1, c29_2, c29_1, c13_1, c30_1, c31_1